Research Article Travelling Wave Solutions for the Coupled...
Transcript of Research Article Travelling Wave Solutions for the Coupled...
Research ArticleTravelling Wave Solutions for the Coupled IBq Equations byUsing the tanh-coth Method
Omer Faruk Gozukizil1 and Samil Akcagil2
1 Department of Mathematics Sakarya University 9054187 Sakarya Turkey2 Vocational School of Pazaryeri Bilecik Seyh Edebali University 9011800 Bilecik Turkey
Correspondence should be addressed to Omer Faruk Gozukizil farukgsakaryaedutr
Received 4 May 2013 Accepted 9 December 2013 Published 19 January 2014
Academic Editor Ch Tsitouras
Copyright copy 2014 O F Gozukizil and S Akcagil This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited
Based on the availability of symbolic computation the tanh-coth method is used to obtain a number of travelling wave solutionsfor several coupled improved Boussinesq equations The abundant new solutions can be seen as improvement of the previouslyknown data The obtained results in this work also demonstrate the efficiency of the method
1 Introduction
We consider the following two coupled improved Boussinesq(IMBq) equations of Sobolev type
119906119905119905minus 1205722119906119909119909119905119905
minus 119906119909119909= 119891(119906 119908)
119909119909 119909 isin R 119905 gt 0
119908119905119905minus 1205722119908119909119909119905119905
minus 119908119909119909= 119892(119906 119908)
119909119909 119909 isin R 119905 gt 0
(1)
where 119891 and 119892 are given nonlinear functions 119906(119909 119905) and119908(119909 119905) are unknown functions and 120572 is a constant that hasbeen derived to describe bidirectional wave propagation inseveral studies for instance in a Toda lattice model with atransversal degree of freedom in a two-layered lattice modeland in a diatomic lattice De Godefroy has studied (1) as theCauchy problem under certain conditions and showed thatthe solution for the Cauchy problem of this system blows upin finite time [1] Wang and Li have considered the Cauchyproblem for (1) proved the existence and uniqueness of theglobal solution and given sufficient conditions of blow-upof the solution in finite time by convex methods [2] TheCauchy problem for (1) has been studied and established theconditions for the global existence and finite-time blow-up ofsolutions in Sobolev spaces119867119904 times119867119904 for 119904 gt 12 [3] For moreinformation we refer the reader to [3] and references therein
Chen and Zhang have considered the initial boundaryvalue problem for the system of the generalized IMBq typeequations
119906119905119905minus 119860119906119909119909minus 119861119906119909119909119905119905
= 119898(119908)119909119909 0 lt 119909 lt 119897
1 119905 gt 0
119908119905119905minus 119861119908119909119909119905119905
= 119899(119908)119909119909 0 lt 119909 lt 119897
1 119905 gt 0
(2)
where 119906(119909 119905) and 119908(119909 119905) are unknown functions 119860 119861 gt 0are constants and119898 and 119899 are the given nonlinear functionsAs a result they have proved the existence and uniquenessof the global generalized solution and the global classicalsolution [4]
Rosenau is concerned with the problem of how todescribe the dynamics of a dense lattice via the system
119906119905119905= (119906 + 2120572119906119908)
119909119909+
1
12
119906119909119909119905119905
119908119905119905= (2119908 + 120573119908
2+ 1205721199062)119909119909+
1
12
119908119909119909119905119905
(3)
where 120572 120573 gt 0 are constants and pointed out that (3) was aconvenient vehicle to study the dynamics of a dense lattice [5]
In [6] a transversal degree of freedom was introducedin the Toda lattice For different order of magnitude of thelongitudinal and transversal strains coupled and uncoupled
Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 486269 14 pageshttpdxdoiorg1011552014486269
2 Journal of Applied Mathematics
equations for these fields were derived in the discrete case aswell as the continuum limit The system
120588
120572
119906119905119905= 120573119906119909119909minus
1205732
2
(1199062)119909119909+
120573
2
(1199082)119909119909+
120588
120572
1198972
12
119906119909119909119905119905
120588
120572
119908119905119905= 120573(119906119908)
119909119909+
120588
120572
1198972
12
119908119909119909119905119905
(4)
has been obtained for the longitudinal and transversal strains119897 and 1119887 are characteristic lengths in the model 120573 is theirratio (ie120573 = 119897119887)120588denotes the linearmass density and120572 gt 0is a constant Besides travellingwave solutions andnumericalsolutions of the system have been found
Turitsyn has proved the existence of blow-up for thecontinuum limit model of the Toda lattice with a transversaldegree of freedom analytically and analysed the coupled IBqequations
119906119905119905= 120573119906119909119909minus
120573
2
(1199062)119909119909+
1
2
(1199082)119909119909+ 1205722119906119909119909119905119905
119908119905119905= (119906119908)
119909119909+ 1205722119908119909119909119905119905
(5)
Turitsyn has showed that the sufficient criterion for thecollapse in this model may be formulated as the requirementof nonpositively of the system Hamiltonian [7]
Pego et al have studied the stability of solitary waves oftwo coupled improved Boussinesq equations
119906119905119905= 119906119909119909minus 120573(119906
2)119909119909+ (1199082)119909119909+ 119906119909119909119905119905
119908119905119905= 119908119909119909+ 2(119906119908)
119909119909+ 119908119909119909119905119905
(6)
which model weakly nonlinear vibrations in a cubic latticeThey have encountered this system for which the classicmethod fails [8]
Fan and Tian have expanded themodel in [6] and consid-ered the following general Toda dense lattice equation whichis governed by the following coupled improved Boussinesqequation
119906119905119905= 120573119906119909119909minus
120573
2
(1199062)119909119909+
120573
2
(1199082)119909119909+ 1205722119906119909119909119905119905
119908119905119905= 120574(119906119908)
119909119909+ 120588(119908
2)119909119909+ 1205722119908119909119909119905119905
(7)
They have studied compacton solutions andmulticompactonsolutions of transversal degree of freedom by direct sine andcosine method and found peakon solutions Moreover theyhave presented more solitary wave solutions for a specialcase of (7) using improved sin-cos and improved sinh-coshmethod [9]
The paper is arranged as followsWe described the outlineof the tanh-coth method in the following section and derivedvarious exact travelling wave solutions of the coupled IBqequations (3) (4) (5) (6) and (7) in the next sections Finallywe summarized our conclusions in the last section
2 Outline of the tanh-coth Method
Wazwaz has summarized the tanh-coth method [10] in thefollowing manner
(i) First consider a general form of nonlinear equation
119875 (119906 119906119905 119906119909 119906119909119909 ) = 0 (8)
(ii) To find the traveling wave solution of (8) the wavevariable 120585 = 119909 minus 119881119905 is introduced so that
119906 (119909 119905) = 119880 (120583120585) (9)
Based on this one may use the following changes
120597
120597119905
= minus119881
119889
119889120585
120597
120597119909
= 120583
119889
119889120585
1205972
1205971199092= 1205832 1198892
1198891205852
1205973
1205971199093= 1205833 1198893
1198891205853
(10)
and so on for other derivatives Using (10) changes thePDE (8) to an ODE
119876(1198801198801015840 11988010158401015840 ) = 0 (11)
(iii) If all terms of the resultingODE contain derivatives in120585 then by integrating this equation and by consider-ing the constant of integration to be zero one obtainsa simplified ODE
(iv) A new independent variable
119884 = tanh (120583120585) (12)
is introducedwhich leads to the change of derivatives
119889
119889120585
= 120583 (1 minus 1198842)
119889
119889119884
1198892
1198891205852= minus2120583
2119884 (1 minus 119884
2)
119889
119889119884
+ 1205832(1 minus 119884
2)
2 1198892
1198891198842
1198893
1198891205853= 21205833(1 minus 119884
2) (31198842minus 1)
119889
119889119884
minus 61205833119884(1 minus 119884
2)
2 1198892
1198891198842+ 1205833(1 minus 119884
2)
3 1198893
1198891198843
1198894
1198891205854= minus8120583
4119884 (1 minus 119884
2) (31198842minus 2)
119889
119889119884
+ 41205834(1 minus 119884
2)
2
(91198842minus 2)
1198892
1198891198842
minus 121205834119884(1 minus 119884
2)
3 1198893
1198891198843
+ 1205834(1 minus 119884
2)
4 1198894
1198891198844
(13)
where other derivatives can be derived in a similarmanner
Journal of Applied Mathematics 3
(v) The ansatz of the form
119880 (120583120585) = 119878 (119884) =
119872
sum
119896=0
119886119896119884119896+
119872
sum
119896=1
119887119896119884minus119896 (14)
is introduced where 119872 is a positive integer in mostcases that will be determined If119872 is not an integerthen a transformation formula is used to overcomethis difficulty Substituting (13) and (14) into the ODE(11) yields an equation in powers of 119884
(vi) To determine the parameter 119872 the linear terms ofthe highest order in the resulting equation with thehighest order nonlinear terms are balanced With119872determined one collects all coefficients of powers of119884 in the resulting equation where these coefficientshave to vanish This will give a system of algebraicequations involving the 119886
119896and 119887119896 (119896 = 0 119872) 119881
and 120583 Having determined these parameters knowingthat119872 is a positive integer in most cases and using(14) one obtains an analytic solution in a closed form
3 The Travelling Wave Solutions
31 The First Coupled We first consider the following twocoupled IBq equations
119906119905119905= (119906 + 2120572119906119908)
119909119909+
1
12
119906119909119909119905119905
119908119905119905= (2119908 + 120573119908
2+ 1205721199062)119909119909+
1
12
119908119909119909119905119905
(15)
Using the wave variable 120585 = 119909 minus 119881119905 then by integrating thisequation twice and considering the constants of integrationto be zero the system (15) is carried to a system of ODEs
12 (1198812minus 1)119880 minus 24120572119880119882 minus 119881
211988010158401015840= 0
12 (1198812minus 2)119882 minus 12120573119882
2minus 12120572119880
2minus 119881211988210158401015840= 0
(16)
Balancing 119881211988010158401015840 with 119880119882 and 119881211988210158401015840 with 1198802 in (16) gives
119872+ 2 = 119872 +119873
2119872 = 119873 + 2
(17)
so that119872 = 119873 = 2 We consider solutions in the form
119880 (120583120585) = 119878 (119884) =
2
sum
119896=0
119886119896119884119896+
2
sum
119896=1
119887119896119884minus119896
119882 (120583120585) = 119878 (119884) =
2
sum
119896=0
119888119896119884119896+
2
sum
119896=1
119889119896119884119896
(18)
Substituting (18) into the two components of (16) and col-lecting the coefficients of 119884 give two systems of algebraic
equations for 1198860 1198861 1198862 1198871 1198872 1198880 1198881 119888211988911198892119881 and120583 Solving
these systems together leads to the following sets
1198860= minus
1198862
3
1198862=
3radic120572120573 (1198814minus 31198812+ 2)
41205722
1198880=
120573 + 4120572
8120572 (120572 minus 120573)
1198882=
minus3120573
8120572 (120572 minus 120573)
119881 =
6
radicminus61205832+ 18
120583 =
radic3120573 (minus3120573 + 4120572)
minus3120573 + 4120572
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860= minus2119887
2 119886
2=
91205732(2120572 minus 120573)
102412057231198872(120572 minus 120573)
2
1198872=
3radic120572120573 (1198814minus 31198812+ 2)
161205722
1198880=
minus3120573 + 8120572
16120572 (120572 minus 120573)
1198882=
3120573
32120572 (120572 minus 120573)
1198892= 1198882
119881 =
6
radic241205832+ 18
120583 =
radic120573 (9120573 minus 12120572)
minus6120573 + 8120572
1198861= 1198871= 1198881= 1198891= 0
1198860=
21198872
3
1198862=
91205732(2120572 minus 120573)
102412057231198872(120572 minus 120573)
2
1198872=
3radic120572120573 (1198814minus 31198812+ 2)
161205722
1198880=
minus120573 + 8120572
16120572 (120572 minus 120573)
1198882=
minus3120573
32120572 (120572 minus 120573)
1198892= 1198882
119881 =
6
radicminus241205832+ 18
120583 =
radic3120573 (minus3120573 + 4120572)
minus6120573 + 8120572
1198861= 1198871= 1198881= 1198891= 0
1198860=
minus91205732(2120572 minus 120573)
6412057231198862(120572 minus 120573)
2 119886
2=
3radic120572120573 (1198814minus 31198812+ 2)
41205722
1198880=
minus3120573 + 4120572
8120572 (120572 minus 120573)
1198882=
3120573
8120572 (120572 minus 120573)
119881 =
6
radic61205832+ 18
120583 =
radic120573 (9120573 minus 12120572)
minus3120573 + 4120572
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
(19)
4 Journal of Applied Mathematics
Consequently we obtain the following travelling wave solu-tions
1199061(119909 119905) = minus
radic120572120573 (1198814minus 31198812+ 2)
41205722
+
3radic120572120573 (1198814minus 31198812+ 2)
41205722
tanh2radic3120573 (minus3120573 + 4120572)
minus3120573 + 4120572
times(119909 minus
6
radicminus61205832+ 18
119905)
1199081(119909 119905) =
120573 + 4120572
8120572 (120572 minus 120573)
minus
3120573
8120572 (120572 minus 120573)
tanh2radic3120573 (minus3120573 + 4120572)
minus3120573 + 4120572
times(119909 minus
6
radicminus61205832+ 18
119905)
1199062(119909 119905) = minus
3radic120572120573 (1198814minus 31198812+ 2)
81205722
+
91205732(2120572 minus 120573)
102412057231198872(120572 minus 120573)
2tanh2
radic120573 (9120573 minus 12120572)
minus6120573 + 8120572
times(119909 minus
6
radic241205832+ 18
119905)
+
3radic120572120573 (1198814minus 31198812+ 2)
161205722
coth2radic120573 (9120573 minus 12120572)
minus6120573 + 8120572
times(119909 minus
6
radic241205832+ 18
119905)
1199082(119909 119905) =
minus3120573 + 8120572
16120572 (120572 minus 120573)
+
3120573
32120572 (120572 minus 120573)
tanh2radic120573 (9120573 minus 12120572)
minus6120573 + 8120572
times(119909 minus
6
radic241205832+ 18
119905)
+
3120573
32120572 (120572 minus 120573)
coth2radic120573 (9120573 minus 12120572)
minus6120573 + 8120572
times(119909 minus
6
radic241205832+ 18
119905)
1199063(119909 119905) =
radic120572120573 (1198814minus 31198812+ 2)
81205722
+
91205732(2120572 minus 120573)
102412057231198872(120572 minus 120573)
2tanh2
radic3120573 (minus3120573 + 4120572)
minus6120573 + 8120572
times(119909 minus
6
radicminus241205832+ 18
119905)
+
3radic120572120573 (1198814minus 31198812+ 2)
161205722
coth2radic3120573 (minus3120573 + 4120572)
minus6120573 + 8120572
times(119909 minus
6
radicminus241205832+ 18
119905)
1199083(119909 119905) =
minus120573 + 8120572
16120572 (120572 minus 120573)
minus
3120573
32120572 (120572 minus 120573)
tanh2radic3120573 (minus3120573 + 4120572)
minus6120573 + 8120572
times(119909 minus
6
radicminus241205832+ 18
119905)
minus
3120573
32120572 (120572 minus 120573)
coth2radic3120573 (minus3120573 + 4120572)
minus6120573 + 8120572
times(119909 minus
6
radicminus241205832+ 18
119905)
1199064(119909 119905) =
minus91205732(2120572 minus 120573)
6412057231198862(120572 minus 120573)
2
+
3radic120572120573 (1198814minus 31198812+ 2)
41205722
tanh2radic120573 (9120573 minus 12120572)
minus3120573 + 4120572
times(119909 minus
6
radic61205832+ 18
119905)
Journal of Applied Mathematics 5
1199084(119909 119905) =
minus3120573 + 4120572
8120572 (120572 minus 120573)
+
3120573
8120572 (120572 minus 120573)
tanh2radic120573 (9120573 minus 12120572)
minus3120573 + 4120572
times(119909 minus
6
radic61205832+ 18
119905)
(20)
32 The Second Coupled We next consider the system ofequations
120588
120572
119906119905119905= 120573119906119909119909minus
1205732
2
(1199062)119909119909+
120573
2
(1199082)119909119909+
120588
120572
1198972
12
119906119909119909119905119905
120588
120572
119908119905119905= 120573(119906119908)
119909119909+
120588
120572
1198972
12
119908119909119909119905119905
(21)
Using thewave variable 120585 = 119909minus119881119905 and then by integrating thisequation twice and considering the constants of integration tobe zero the system (21) is carried to a system of ODEs
12 (120572120573 minus 1205881198812)119880 minus 6120572120573
21198802+ 6120572120573119882
2+ 1205881198972119881211988010158401015840= 0
12120572120573119880119882 + 1205881198972119881211988210158401015840minus 12120588119881
2119882 = 0
(22)
Balancing minus612057212057321198802 with 120588119897
2119881211988010158401015840 and 12120572120573119880119882 with
1205881198972119881211988210158401015840 in (22) gives
2119872 = 119872 + 2
2119873 = 119872 +119873
(23)
so that119872 = 119873 = 2 The tanh-coth method admits the use of
119880 (120583120585) = 119878 (119884) =
2
sum
119896=0
119886119896119884119896+
2
sum
119896=1
119887119896119884minus119896
119882 (120583120585) = 119878 (119884) =
2
sum
119896=0
119888119896119884119896+
2
sum
119896=1
119889119896119884119896
(24)
Substituting (24) into (22) collecting the coefficients of 119884and solving the resulting system we find the following setsof solutions
1198860=
1
120573 + 1
1198862=
minus1
2 (120573 + 1)
1198872= 1198862
1198892= plusmn
radic120573 + 2
2 (120573 + 1)
1198882= minus1198892
119881 =
radic3120588120572120573 (120573 + 1)
3120588 (120573 + 1)
120583 =
radic3
119897
1198861= 1198871= 1198880= 1198881= 1198891= 0
1198860=
1
120573 + 1
1198862=
3120573
61205732+ 14120573 + 8
1198872= 1198862
1198892= plusmn
3120573radic120573 + 2
2 (3120573 + 4) (120573 + 1)
1198882= minus1198892
119881 =
radic15120588120572120573 (211989721205832+ 9120573 + 15)
120588 (211989721205832+ 9120573 + 15)
120583 =
radicminus120573 (15120573 + 12)
(5120573 + 4) 119897
1198861= 1198871= 1198880= 1198881= 1198891= 0
1198860=
4 minus 120573
4 (120573 + 1)
1198862=
3120573
4120573 + 1
1198880=
minus1198882
3
1198882= plusmn
9120573
2radic611989721205832+ 36120573 + 18
119881 =
radicminus3120588120572120573 (minus3 + 11989721205832)
120588 (minus3 + 11989721205832)
120583 =
radicminus120573 (3120573 + 6)
(120573 + 2) 119897
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860=
3120573 + 4
4 (120573 + 1)
1198862=
minus3120573
4120573 + 1
1198880= minus1198882
1198882= plusmn
9120573
2radicminus611989721205832+ 36120573 + 18
119881 =
radic3120588120572120573 (3 + 11989721205832)
120588 (3 + 11989721205832)
120583 =
radic3120573 (120573 + 2)
(120573 + 2) 119897
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860=
120573 + 8
8 (120573 + 1)
1198862=
3120573
16 (120573 + 1)
1198872= 1198862 119888
0=
21198892
3
1198882= 1198892 119889
2= plusmn
9120573
8radic2411989721205832+ 36120573 + 18
119881 =
radicminus3120588120572120573 (minus3 + 411989721205832)
120588 (minus3 + 411989721205832)
120583 =
radicminus120573 (3120573 + 6)
2 (120573 + 2) 119897
1198861= 1198871= 1198881= 1198891= 0
1198860=
3 (120573 + 8)
8 (120573 + 1)
1198862=
minus3120573
16 (120573 + 1)
1198872= 1198862 119888
0= minus2119889
2
6 Journal of Applied Mathematics
1198882= 1198892 119889
2= plusmn
9120573
8radicminus2411989721205832+ 36120573 + 18
119881 =
radic3120588120572120573 (3 + 411989721205832)
120588 (3 + 411989721205832)
120583 =
radic120573 (3120573 + 6)
2 (120573 + 2) 119897
1198861= 1198871= 1198881= 1198891= 0
(25)
In accordance with these sets of solutions we have thefollowing solutions
1199061(119909 119905) =
1
120573 + 1
minus
1
2 (120573 + 1)
tanh2radic3
119897
times (119909 minus
radic3120588120572120573 (120573 + 1)
3120588 (120573 + 1)
119905)
minus
1
2 (120573 + 1)
coth2radic3
119897
(119909 minus
radic3120588120572120573 (120573 + 1)
3120588 (120573 + 1)
119905)
1199081(119909 119905) = ∓
radic120573 + 2
2 (120573 + 1)
tanh2radic3
119897
(119909 minus
radic3120588120572120573 (120573 + 1)
3120588 (120573 + 1)
119905)
plusmn
radic120573 + 2
2 (120573 + 1)
coth2radic3
119897
(119909 minus
radic3120588120572120573 (120573 + 1)
3120588 (120573 + 1)
119905)
1199062(119909 119905) =
1
120573 + 1
+
3120573
61205732+ 14120573 + 8
tanh2radicminus120573 (15120573 + 12)
(5120573 + 4) 119897
times (119909 minus
radic15120588120572120573 (211989721205832+ 9120573 + 15)
120588 (211989721205832+ 9120573 + 15)
119905)
+
3120573
61205732+ 14120573 + 8
coth2radicminus120573 (15120573 + 12)
(5120573 + 4) 119897
times (119909 minus
radic15120588120572120573 (211989721205832+ 9120573 + 15)
120588 (211989721205832+ 9120573 + 15)
119905)
1199082(119909 119905) = ∓
3120573radic120573 + 2
2 (3120573 + 4) (120573 + 1)
tanh2radicminus120573 (15120573 + 12)
(5120573 + 4) 119897
times (119909 minus
radic15120588120572120573 (211989721205832+ 9120573 + 15)
120588 (211989721205832+ 9120573 + 15)
119905)
plusmn
3120573radic120573 + 2
2 (3120573 + 4) (120573 + 1)
coth2radicminus120573 (15120573 + 12)
(5120573 + 4) 119897
times (119909 minus
radic15120588120572120573 (211989721205832+ 9120573 + 15)
120588 (211989721205832+ 9120573 + 15)
119905)
1199063(119909 119905) =
4 minus 120573
4 (120573 + 1)
+
3120573
4120573 + 1
tanh2radicminus120573 (3120573 + 6)
(120573 + 2) 119897
times (119909 minus
radicminus3120588120572120573 (minus3 + 11989721205832)
120588 (minus3 + 11989721205832)
119905)
1199083(119909 119905) = ∓
3120573
2radic611989721205832+ 36120573 + 18
plusmn
9120573
2radic611989721205832+ 36120573 + 18
tanh2radicminus120573 (3120573 + 6)
(120573 + 2) 119897
times (119909 minus
radicminus3120588120572120573 (minus3 + 11989721205832)
120588 (minus3 + 11989721205832)
119905)
1199064(119909 119905) =
3120573 + 4
4 (120573 + 1)
minus
3120573
4120573 + 1
tanh2radic3120573 (120573 + 2)
(120573 + 2) 119897
times (119909 minus
radic3120588120572120573 (3 + 11989721205832)
120588 (3 + 11989721205832)
119905)
1199084(119909 119905) = ∓
9120573
2radicminus611989721205832+ 36120573 + 18
plusmn
9120573
2radicminus611989721205832+ 36120573 + 18
tanh2radic3120573 (120573 + 2)
(120573 + 2) 119897
times (119909 minus
radic3120588120572120573 (3 + 11989721205832)
120588 (3 + 11989721205832)
119905)
1199065(119909 119905) =
120573 + 8
8 (120573 + 1)
+
3120573
16 (120573 + 1)
tanh2radicminus120573 (3120573 + 6)
2 (120573 + 2) 119897
Journal of Applied Mathematics 7
times(119909 minus
radicminus3120588120572120573 (minus3 + 411989721205832)
120588 (minus3 + 411989721205832)
119905)
+
3120573
16 (120573 + 1)
coth2radicminus120573 (3120573 + 6)
2 (120573 + 2) 119897
times (119909 minus
radicminus3120588120572120573 (minus3 + 411989721205832)
120588 (minus3 + 411989721205832)
119905)
1199085(119909 119905) = plusmn
3120573
4radic2411989721205832+ 36120573 + 18
plusmn
9120573
8radic2411989721205832+ 36120573 + 18
tanh2radicminus120573 (3120573 + 6)
2 (120573 + 2) 119897
times (119909 minus
radicminus3120588120572120573 (minus3 + 411989721205832)
120588 (minus3 + 411989721205832)
119905)
plusmn
9120573
8radic2411989721205832+ 36120573 + 18
coth2radicminus120573 (3120573 + 6)
2 (120573 + 2) 119897
times (119909 minus
radicminus3120588120572120573 (minus3 + 411989721205832)
120588 (minus3 + 411989721205832)
119905)
1199066(119909 119905) =
3 (120573 + 8)
8 (120573 + 1)
minus
3120573
16 (120573 + 1)
tanh2radic120573 (3120573 + 6)
2 (120573 + 2) 119897
times (119909 minus
radic3120588120572120573 (3 + 411989721205832)
120588 (3 + 411989721205832)
119905)
minus
3120573
16 (120573 + 1)
coth2radic120573 (3120573 + 6)
2 (120573 + 2) 119897
times (119909 minus
radic3120588120572120573 (3 + 411989721205832)
120588 (3 + 411989721205832)
119905)
1199086(119909 119905) = ∓
9120573
4radicminus2411989721205832+ 36120573 + 18
plusmn
9120573
8radicminus2411989721205832+ 36120573 + 18
tanh2radic120573 (3120573 + 6)
2 (120573 + 2) 119897
times (119909 minus
radic3120588120572120573 (3 + 411989721205832)
120588 (3 + 411989721205832)
119905)
plusmn
9120573
8radicminus2411989721205832+ 36120573 + 18
coth2radic120573 (3120573 + 6)
2 (120573 + 2) 119897
times (119909 minus
radic3120588120572120573 (3 + 411989721205832)
120588 (3 + 411989721205832)
119905)
(26)
33 The Third Coupled We now examine the coupled equa-tions
119906119905119905= 120573119906119909119909minus
120573
2
(1199062)119909119909+
1
2
(1199082)119909119909+ 1205722119906119909119909119905119905
119908119905119905= (119906119908)
119909119909+ 1205722119908119909119909119905119905
(27)
Using the wave variable 120585 = 119909 minus119881119905 the system (27) is carriedto a system of ODEs Integrating these equations twice andneglecting constants of integration we find
2 (1198812minus 120573)119880 + 120573119880
2minus 120573119882
2minus 21205722119881211988010158401015840= 0
1198812119882minus119880119882 minus 120572
2119881211988210158401015840= 0
(28)
Balancing 1205731198822 with 21205722119881211988010158401015840 and 119881211988210158401015840 with 119880119882 in (28)gives
2119873 = 119872 + 2
2 + 119873 = 119872 +119873
(29)
so that
119872 = 119873 = 2 (30)
As a result we seek solutions to (27) in the form
119880(120583120585) = 119878 (119884) =
2
sum
119896=0
119886119896119884119896+
2
sum
119896=1
119887119896119884minus119896
119882 (120583120585) = 119878 (119884) =
2
sum
119896=0
119888119896119884119896+
2
sum
119896=1
119889119896119884119896
(31)
Substitution of (31) into (28) leads to an algebraic system inthe unknowns 119886
0 1198861 1198862 1198871 1198872 1198880 1198881 1198882 1198891 1198892 119881 and 120583
8 Journal of Applied Mathematics
Collecting the coefficients of 119884 and solving the resultingsystem we find the following sets of solutions
1198860=
120573 (3120573 + 8)
8 (120573 + 1)
1198862=
minus31205732
16 (120573 + 1)
1198872= 1198862 119888
0= minus2119889
2 119888
2= 1198892
1198892=
plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573
16
119881 =
radic120573 (1612057221205832+ 1)
1612057221205832+ 1
120583 =
radic120573 (120573 + 2)
4120572 (120573 + 2)
1198861= 1198871= 1198881= 1198891= 0
1198860=
120573 (120573 + 8)
8 (120573 + 1)
1198862=
31205732
16 (120573 + 1)
1198872= 1198862
1198880=
21198892
3
1198882= 1198892
1198892=
plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573
16
119881 =
radicminus120573 (1612057221205832minus 1)
1612057221205832minus 1
120583 =
radicminus120573 (120573 + 2)
4120572 (120573 + 2)
1198861= 1198871= 1198881= 1198891= 0
1198860=
minus120573 (120573 minus 4)
4 (120573 + 1)
1198862=
31205732
4 (120573 + 1)
1198880= minus
1198882
3
1198882=
plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573
4
119881 =
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
120583 =
radicminus120573 (120573 + 2)
2120572 (120573 + 2)
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860=
120573 (3120573 + 4)
4 (120573 + 1)
1198862=
minus31205732
4 (120573 + 1)
1198880= minus2
1198882=
plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573
4
119881 =
radic120573 (412057221205832+ 1)
412057221205832+ 1
120583 =
radic120573 (120573 + 2)
2120572 (120573 + 2)
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860=
120573
120573 + 1
1198862=
31205732
2 (31205732+ 7120573 + 4)
1198872= 1198862 119888
2= minus1198892
1198892=
plusmn3radicminus421198814120573 minus 58119881
4minus 21198812120573 minus 120119881
2+ 120120573
8
119881 =
radic5120573 (812057221205832+ 3120573 + 5)
812057221205832+ 3120573 + 5
120583 =
radicminus120573 (5120573 + 4)
2120572 (5120573 + 4)
1198861= 1198871= 1198880= 1198881= 1198891= 0
1198860=
120573
120573 + 1
1198862=
minus120573
2 (120573 + 1)
1198872= 1198862
1198882= minus1198892 119889
2=
plusmnradic91198812+ 6
2
119881 =
radic3120573 (120573 + 1)
3 (120573 + 1)
120583 =
minus1
2120572
1198861= 1198871= 1198880= 1198881= 1198891= 0
(32)
Consequently we obtain the following solutions
1199061(119909 119905) =
120573 (3120573 + 8)
8 (120573 + 1)
minus
31205732
16 (120573 + 1)
tanh2radic120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radic120573 (1612057221205832+ 1)
1612057221205832+ 1
119905)
minus
31205732
16 (120573 + 1)
coth2radic120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radic120573 (1612057221205832+ 1)
1612057221205832+ 1
119905)
1199081(119909 119905) =
∓3radicminus41198814minus 41198812+ 21205732+ 4120573
8
plusmn
3radicminus41198814minus 41198812+ 21205732+ 4120573
16
tanh2radic120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radic120573 (1612057221205832+ 1)
1612057221205832+ 1
119905)
plusmn
3radicminus41198814minus 41198812+ 21205732+ 4120573
16
coth2radic120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radic120573 (1612057221205832+ 1)
1612057221205832+ 1
119905)
Journal of Applied Mathematics 9
1199062(119909 119905) =
120573 (120573 + 8)
8 (120573 + 1)
+
31205732
16 (120573 + 1)
tanh2radicminus120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radicminus120573 (1612057221205832minus 1)
1612057221205832minus 1
119905)
+
31205732
16 (120573 + 1)
coth2radicminus120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radicminus120573 (1612057221205832minus 1)
1612057221205832minus 1
119905)
1199082(119909 119905) = plusmn
radicminus41198814minus 41198812+ 21205732+ 4120573
8
plusmn
3radicminus41198814minus 41198812+ 21205732+ 4120573
16
tanh2radicminus120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radicminus120573 (1612057221205832minus 1)
1612057221205832minus 1
119905)
plusmn
3radicminus41198814minus 41198812+ 21205732+ 4120573
16
coth2radicminus120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radicminus120573 (1612057221205832minus 1)
1612057221205832minus 1
119905)
1199063(119909 119905) =
minus120573 (120573 minus 4)
4 (120573 + 1)
+
31205732
4 (120573 + 1)
tanh2radicminus120573 (120573 + 2)
2120572 (120573 + 2)
times (119909 minus
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
119905)
1199083(119909 119905) = ∓
radicminus41198814minus 41198812+ 21205732+ 4120573
4
plusmn
3radicminus41198814minus 41198812+ 21205732+ 4120573
4
tanhradicminus120573 (120573 + 2)
2120572 (120573 + 2)
times (119909 minus
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
119905)
1199064(119909 119905) =
120573 (3120573 + 4)
4 (120573 + 1)
minus
31205732
4 (120573 + 1)
tanh2radic120573 (120573 + 2)
2120572 (120573 + 2)
times (119909 minus
radic120573 (412057221205832+ 1)
412057221205832+ 1
119905)
1199084(119909 119905) = ∓
3radicminus41198814minus 41198812+ 21205732+ 4120573
4
plusmn
3radicminus41198814minus 41198812+ 21205732+ 4120573
4
tanh2radic120573 (120573 + 2)
2120572 (120573 + 2)
times (119909 minus
radic120573 (412057221205832+ 1)
412057221205832+ 1
119905)
1199065(119909 119905) =
120573
120573 + 1
+
31205732
2 (31205732+ 7120573 + 4)
tanh2radicminus120573 (5120573 + 4)
2120572 (5120573 + 4)
times (119909 minus
radic5120573 (812057221205832+ 3120573 + 5)
812057221205832+ 3120573 + 5
119905)
+
31205732
2 (31205732+ 7120573 + 4)
coth2radicminus120573 (5120573 + 4)
2120572 (5120573 + 4)
times (119909 minus
radic5120573 (812057221205832+ 3120573 + 5)
812057221205832+ 3120573 + 5
119905)
1199085(119909 119905)
= ∓
3radicminus421198814120573 minus 58119881
4minus 21198812120573 minus 120119881
2+ 120120573
8
times tanh2radicminus120573 (5120573 + 4)
2120572 (5120573 + 4)
times (119909 minus
radic5120573 (812057221205832+ 3120573 + 5)
812057221205832+ 3120573 + 5
119905)
plusmn
3radicminus421198814120573 minus 58119881
4minus 21198812120573 minus 120119881
2+ 120120573
8
10 Journal of Applied Mathematics
times coth2radicminus120573 (5120573 + 4)
2120572 (5120573 + 4)
times (119909 minus
radic5120573 (812057221205832+ 3120573 + 5)
812057221205832+ 3120573 + 5
119905)
1199066(119909 119905) =
120573
120573 + 1
minus
120573
2 (120573 + 1)
tanh2minus12120572
(119909 minus
radic3120573 (120573 + 1)
3 (120573 + 1)
119905)
minus
120573
2 (120573 + 1)
coth2minus12120572
(119909 minus
radic3120573 (120573 + 1)
3 (120573 + 1)
119905)
1199086(119909 119905)
= ∓
radic91198812+ 6
2
tanh2minus12120572
(119909 minus
radic3120573 (120573 + 1)
3 (120573 + 1)
119905)
plusmn
radic91198812+ 6
2
coth2minus12120572
(119909 minus
radic3120573 (120573 + 1)
3 (120573 + 1)
119905)
(33)
34 The Fourth Coupled We now consider the coupled IBqequations
119906119905119905= 119906119909119909minus 120573(119906
2)119909119909+ (1199082)119909119909+ 119906119909119909119905119905
119908119905119905= 119908119909119909+ 2(119906119908)
119909119909+ 119908119909119909119905119905
(34)
Using the wave variable 120585 = 119909 minus 119881119905 (34) can be converted toa system of ODEs
(1198812minus 1)119880 + 120573119880
2minus1198822minus 119881211988010158401015840= 0
(1198812minus 1)119882 minus 2119880119882 minus 119881
211988210158401015840= 0
(35)
Balancing1198822 with 11988010158401015840 and 119880119882 with11988210158401015840 in (35) gives
2119873 = 119872 + 2
119872 + 119873 = 119873 + 2
(36)
so that
119872 = 119873 = 2 (37)
The tanh-coth method substitutes the finite expansions
119880(120583120585) = 119878 (119884) =
2
sum
119896=0
119886119896119884119896+
2
sum
119896=1
119887119896119884minus119896
119882 (120583120585) = 119878 (119884) =
2
sum
119896=0
119888119896119884119896+
2
sum
119896=1
119889119896119884119896
(38)
In (35) collecting the coefficients of each power of 119884 andsolving the resulting system we obtain
1198860=
minus21205832
161205832+ 1
1198862=
minus31205832
161205832+ 1
1198862= 1198872
1198880=
21198892
3
1198882= 1198892
1198892=
plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
119881 =
1
radic161205832+ 1
1198861= 1198871= 1198881= 1198891= 0
1198860=
minus61205832
161205832minus 1
1198862=
31205832
161205832minus 1
1198862= 1198872
1198880= minus2119889
2 119888
2= 1198892
1198892=
plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
119881 =
1
radicminus161205832+ 1
1198861= 1198871= 1198881= 1198891= 0
1198860=
1205832
41205832+ 1
1198862=
minus31205832
41205832+ 1
1198880=
minus1198882
3
1198882=
plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
119881 =
1
radic41205832+ 1
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860=
minus31205832
41205832minus 1
1198862=
31205832
41205832minus 1
1198880= minus1198882
1198882=
plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
119881 =
1
radicminus41205832+ 1
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860=
minus31205832
41205832minus 1
1198872=
31205832
41205832minus 1
1198880= minus1198892
1198892=
plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
119881 =
1
radicminus41205832+ 1
1198861= 1198862= 1198871= 1198881= 1198882= 1198891= 0
(39)
Journal of Applied Mathematics 11
where 120583 is left as a free parameter Consequently we obtainthe following travelling wave solutions
1199061(119909 119905) = minus
21205832
161205832+ 1
minus
31205832
161205832+ 1
tanh2120583(119909 minus 1
radic161205832+ 1
119905)
minus
31205832
161205832+ 1
coth2120583(119909 minus 1
radic161205832+ 1
119905)
1199081(119909 119905) = plusmn
(119881 + 1) (119881 minus 1)radic120573 + 2
8
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
tanh2120583
times(119909 minus
1
radic161205832+ 1
119905)
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
coth2120583
times(119909 minus
1
radic161205832+ 1
119905)
1199062(119909 119905) = minus
61205832
161205832minus 1
+
31205832
161205832minus 1
tanh2120583(119909 minus 1
radicminus161205832+ 1
119905)
+
31205832
161205832minus 1
coth2120583(119909 minus 1
radicminus161205832+ 1
119905)
1199082(119909 119905) = ∓
3 (119881 + 1) (119881 minus 1)radic120573 + 2
8
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
tanh2120583
times(119909 minus
1
radicminus161205832+ 1
119905)
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
coth2120583
times(119909 minus
1
radicminus161205832+ 1
119905)
1199063(119909 119905) =
1205832
41205832+ 1
minus
31205832
41205832+ 1
tanh2120583(119909 minus 1
radic41205832+ 1
119905)
1199083(119909 119905) = ∓
(119881 + 1) (119881 minus 1)radic120573 + 2
4
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
times tanh2120583(119909 minus 1
radic41205832+ 1
119905)
1199064(119909 119905) = minus
31205832
41205832minus 1
+
31205832
41205832minus 1
tanh2120583(119909 minus 1
radicminus41205832+ 1
119905)
1199084(119909 119905) = ∓
3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
times tanh2120583(119909 minus 1
radicminus41205832+ 1
119905)
1199065(119909 119905) = minus
31205832
41205832minus 1
+
31205832
41205832minus 1
coth2120583(119909 minus 1
radicminus41205832+ 1
119905)
1199085(119909 119905) = ∓
3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
plusmn 3
(119881 + 1) (119881 minus 1)radic120573 + 2
4
times coth2120583(119909 minus 1
radicminus41205832+ 1
119905)
(40)
35 The Fifth Coupled We finally consider the coupled IBqequations
119906119905119905= 120573119906119909119909minus
120573
2
(1199062)119909119909+
120573
2
(1199082)119909119909+ 1205722119906119909119909119905119905
119908119905119905= 120574(119906119908)
119909119909+ 120588(119908
2)119909119909+ 1205722119908119909119909119905119905
(41)
12 Journal of Applied Mathematics
This system of equations (41) can be converted to the follow-ing system of ODEs by using the wave variable 120585 = 119909 minus 119881119905
(21198812minus 2120573)119880 + 120573119880
2minus 120573119882
2minus 21205722119881211988010158401015840= 0
1198812119882minus 120574119880119882 minus 120588119882
2minus 1205722119881211988210158401015840= 0
(42)
Using the balancing procedure in (42) gives
2119872 = 119872 +119873
2119873 = 119872 +119873
(43)
so that
119872 = 119873 = 2 (44)
We consider solutions in the following form
119880 (120583120585) = 119878 (119884) =
2
sum
119896=0
119886119896119884119896+
2
sum
119896=1
119887119896119884minus119896
119882 (120583120585) = 119878 (119884) =
2
sum
119896=0
119888119896119884119896+
2
sum
119896=1
119889119896119884119896
(45)
Substituting (45) into (42) collecting the coefficients of 119884and solving the resulting system we find the following setsof solutions
1198860= (minus120573 (8120588119888
21205732120574 minus 8120573120574
21205881198882minus 18120588
21205732+ 24120573120588
31198882
+1212057321205742minus 16120574
31205881198882+ 24120588
31198882120574 minus 3120574120573
3))
times (4 (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882+ 312058821205733
+ 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882minus 312057331205742minus 312057431205732))
minus1
1198862=
minus (41205881198882120573 + 4119888
2120588120574 minus 3120573
2)
4120574 (120583 + 120573)
1198880= (120573(120588
212057321198882+ 120574211988821205732+ 12057311988821205743+ 2120574212058821198882
+312057412058821205731198882minus
3
2
1205881205732120574 minus
3
4
1205881205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882+ 312058821205733
+ 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882minus 312057331205742minus 312057431205732)
minus1
1198882= (plusmn3 (minus 120573120588 + (minus120588
21205732+ 41205743120573 minus 4120574
21198814
minus 412057431198812+ 212057321205742
minus41205882120574120573 + 4120588
21198814+ 412058821198812120574)
12
))
times (4 (1205742minus 1205882))
minus1
119881 =
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
120583 =
radicminus120573 (2120574 + 120573)
2120572 (2120574 + 120573)
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860= (120573(6120588119888
21205732120574 + 10120573120574
21205881198882+
3
2
12058821205732+ 2120573120588
31198882
+312057321205742+ 412057431205881198882+ 212058831198882120574 +
9
4
1205741205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882minus 312058821205733
minus 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882+ 312057331205742+ 312057431205732)
minus1
1198862=
minus (41205881198882120573 + 4119888
2120588120574 + 3120573
2)
4120574 (120583 + 120573)
1198880= (120573(minus 3120588
212057321198882minus 3120574211988821205732minus 312057311988821205743
minus6120574212058821198882minus 9120574120588
21205731198882minus
9
2
1205881205732120574 minus
9
4
1205881205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882
minus 312058821205733minus 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882+ 312057331205742+ 312057431205732)
minus1
1198882= (plusmn3 (120573120588 + (minus120588
21205732+ 41205743120573 minus 4120574
21198814
minus 412057431198812+ 212057321205742minus 41205882120574120573
+412058821198814+ 412058821198812120574)
12
))
times (4 (1205742minus 1205882))
minus1
119881 =
radic120573 (412057221205832+ 1)
412057221205832+ 1
120583 =
radic120573 (2120574 + 120573)
2120572 (2120574 + 120573)
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
(46)
Journal of Applied Mathematics 13
In view of this we obtain the following travelling wavesolutions
1199061(119909 119905) = (minus120573 (8120588119888
21205732120574 minus 8120573120574
21205881198882minus 18120588
21205732+ 24120573120588
31198882
+1212057321205742minus 16120574
31205881198882+ 24120588
31198882120574 minus 3120574120573
3))
times (4 (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882+ 312058821205733
+ 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882minus 312057331205742minus 312057431205732))
minus1
minus
(41205881198882120573 + 4119888
2120588120574 minus 3120573
2)
4120574 (120583 + 120573)
tanh2radicminus120573 (2120574 + 120573)
2120572 (2120574 + 120573)
times(119909 minus
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
119905)
1199081(119909 119905) = (120573 (120588
212057321198882+ 120574211988821205732+ 12057311988821205743+ 2120574212058821198882
+312057412058821205731198882minus
3
2
1205881205732120574 minus
3
4
1205881205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882+ 312058821205733
+ 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882minus 312057331205742minus 312057431205732)
minus1
plusmn (3 (minus 120573120588 + (minus12058821205732+ 41205743120573 minus 4120574
21198814
minus 412057431198812+ 212057321205742minus 41205882120574120573
+412058821198814+ 412058821198812120574)
12
))
times (4 (1205742minus 1205882))
minus1
times tanh2radicminus120573 (2120574 + 120573)
2120572 (2120574 + 120573)
times(119909 minus
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
119905)
1199062(119909 119905) = (120573 (6120588119888
21205732120574 + 10120573120574
21205881198882+
3
2
12058821205732+ 2120573120588
31198882
+312057321205742+ 412057431205881198882+ 212058831198882120574 +
9
4
1205741205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882minus 312058821205733
minus 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882+ 312057331205742+ 312057431205732)
minus1
minus
(41205881198882120573 + 4119888
2120588120574 + 3120573
2)
4120574 (120583 + 120573)
tanh2radic120573 (2120574 + 120573)
2120572 (2120574 + 120573)
times(119909 minus
radic120573 (412057221205832+ 1)
412057221205832+ 1
119905)
1199082(119909 119905) = (120573 (minus 3120588
212057321198882minus 3120574211988821205732minus 312057311988821205743
minus6120574212058821198882minus 9120574120588
21205731198882minus
9
2
1205881205732120574 minus
9
4
1205881205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882minus 312058821205733
minus 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882+ 312057331205742+ 312057431205732)
minus1
plusmn (3 (120573120588 + (minus12058821205732+ 41205743120573 minus 4120574
21198814
minus 412057431198812+ 212057321205742minus 41205882120574120573
+412058821198814+ 412058821198812120574)
12
))
times (4 (1205742minus 1205882))
minus1
times tanh2radic120573 (2120574 + 120573)
2120572 (2120574 + 120573)
(119909 minus
radic120573 (412057221205832+ 1)
412057221205832+ 1
119905)
(47)
4 Conclusion
In summary the tanh-coth method has been successfullyimplemented to find new travelling wave solutions for severalkinds of two coupled improved Boussinesq equations Thesolutions that have been found can be seen as an expansionand verification of the previously known data However thisstudy confirmed again the belief that the tanh-coth methodis a powerful and reliable technique to handle nonlineardispersive-dissipative equations and system of equationsThroughout the work Maple has been used to overcome thetedious algebraic calculations
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] A De Godefroy ldquoBlow up of solutions of a generalized Boussi-nesq equationrdquo IMA Journal of AppliedMathematics vol 60 no2 pp 123ndash138 1998
14 Journal of Applied Mathematics
[2] S Wang and M Li ldquoThe Cauchy problem for coupled IMBqequationsrdquo IMA Journal of Applied Mathematics vol 74 no 5pp 726ndash740 2009
[3] N Duruk H A Erbay and A Erkip ldquoBlow-up and globalexistence for a general class of nonlocal nonlinear coupled waveequationsrdquo Journal of Differential Equations vol 250 no 3 pp1448ndash1459 2011
[4] G Chen and H Zhang ldquoInitial boundary value problem for asystem of generalized IMBq equationsrdquoMathematical Methodsin the Applied Sciences vol 27 no 5 pp 497ndash518 2004
[5] P Rosenau ldquoDynamics of dense latticesrdquo Physical Review B vol36 no 11 pp 5868ndash5876 1987
[6] P L Christiansen P S Lomdahl andVMuto ldquoOn aToda latticemodel with a transversal degree of freedomrdquo Nonlinearity vol4 no 2 pp 477ndash501 1991
[7] S K Turitsyn ldquoOn a Toda lattice model with a transversaldegree of freedom Sufficient criterion of blow-up in thecontinuum limitrdquo Physics Letters A vol 173 no 3 pp 267ndash2691993
[8] R L Pego P Smereka and M I Weinstein ldquoOscillatoryinstability of solitary waves in a continuum model of latticevibrationsrdquo Nonlinearity vol 8 no 6 pp 921ndash941 1995
[9] X Fan and L Tian ldquoCompacton solutions and multiple com-pacton solutions for a continuum Toda lattice modelrdquo ChaosSolitons and Fractals vol 29 no 4 pp 882ndash894 2006
[10] A M Wazwaz ldquoThe Hirotarsquos bilinear method and the tanh-coth method for multiple-soliton solutions of the Sawada-Kotera-Kadomtsev-Petviashvili equationrdquoAppliedMathematicsand Computation vol 200 no 1 pp 160ndash166 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Journal of Applied Mathematics
equations for these fields were derived in the discrete case aswell as the continuum limit The system
120588
120572
119906119905119905= 120573119906119909119909minus
1205732
2
(1199062)119909119909+
120573
2
(1199082)119909119909+
120588
120572
1198972
12
119906119909119909119905119905
120588
120572
119908119905119905= 120573(119906119908)
119909119909+
120588
120572
1198972
12
119908119909119909119905119905
(4)
has been obtained for the longitudinal and transversal strains119897 and 1119887 are characteristic lengths in the model 120573 is theirratio (ie120573 = 119897119887)120588denotes the linearmass density and120572 gt 0is a constant Besides travellingwave solutions andnumericalsolutions of the system have been found
Turitsyn has proved the existence of blow-up for thecontinuum limit model of the Toda lattice with a transversaldegree of freedom analytically and analysed the coupled IBqequations
119906119905119905= 120573119906119909119909minus
120573
2
(1199062)119909119909+
1
2
(1199082)119909119909+ 1205722119906119909119909119905119905
119908119905119905= (119906119908)
119909119909+ 1205722119908119909119909119905119905
(5)
Turitsyn has showed that the sufficient criterion for thecollapse in this model may be formulated as the requirementof nonpositively of the system Hamiltonian [7]
Pego et al have studied the stability of solitary waves oftwo coupled improved Boussinesq equations
119906119905119905= 119906119909119909minus 120573(119906
2)119909119909+ (1199082)119909119909+ 119906119909119909119905119905
119908119905119905= 119908119909119909+ 2(119906119908)
119909119909+ 119908119909119909119905119905
(6)
which model weakly nonlinear vibrations in a cubic latticeThey have encountered this system for which the classicmethod fails [8]
Fan and Tian have expanded themodel in [6] and consid-ered the following general Toda dense lattice equation whichis governed by the following coupled improved Boussinesqequation
119906119905119905= 120573119906119909119909minus
120573
2
(1199062)119909119909+
120573
2
(1199082)119909119909+ 1205722119906119909119909119905119905
119908119905119905= 120574(119906119908)
119909119909+ 120588(119908
2)119909119909+ 1205722119908119909119909119905119905
(7)
They have studied compacton solutions andmulticompactonsolutions of transversal degree of freedom by direct sine andcosine method and found peakon solutions Moreover theyhave presented more solitary wave solutions for a specialcase of (7) using improved sin-cos and improved sinh-coshmethod [9]
The paper is arranged as followsWe described the outlineof the tanh-coth method in the following section and derivedvarious exact travelling wave solutions of the coupled IBqequations (3) (4) (5) (6) and (7) in the next sections Finallywe summarized our conclusions in the last section
2 Outline of the tanh-coth Method
Wazwaz has summarized the tanh-coth method [10] in thefollowing manner
(i) First consider a general form of nonlinear equation
119875 (119906 119906119905 119906119909 119906119909119909 ) = 0 (8)
(ii) To find the traveling wave solution of (8) the wavevariable 120585 = 119909 minus 119881119905 is introduced so that
119906 (119909 119905) = 119880 (120583120585) (9)
Based on this one may use the following changes
120597
120597119905
= minus119881
119889
119889120585
120597
120597119909
= 120583
119889
119889120585
1205972
1205971199092= 1205832 1198892
1198891205852
1205973
1205971199093= 1205833 1198893
1198891205853
(10)
and so on for other derivatives Using (10) changes thePDE (8) to an ODE
119876(1198801198801015840 11988010158401015840 ) = 0 (11)
(iii) If all terms of the resultingODE contain derivatives in120585 then by integrating this equation and by consider-ing the constant of integration to be zero one obtainsa simplified ODE
(iv) A new independent variable
119884 = tanh (120583120585) (12)
is introducedwhich leads to the change of derivatives
119889
119889120585
= 120583 (1 minus 1198842)
119889
119889119884
1198892
1198891205852= minus2120583
2119884 (1 minus 119884
2)
119889
119889119884
+ 1205832(1 minus 119884
2)
2 1198892
1198891198842
1198893
1198891205853= 21205833(1 minus 119884
2) (31198842minus 1)
119889
119889119884
minus 61205833119884(1 minus 119884
2)
2 1198892
1198891198842+ 1205833(1 minus 119884
2)
3 1198893
1198891198843
1198894
1198891205854= minus8120583
4119884 (1 minus 119884
2) (31198842minus 2)
119889
119889119884
+ 41205834(1 minus 119884
2)
2
(91198842minus 2)
1198892
1198891198842
minus 121205834119884(1 minus 119884
2)
3 1198893
1198891198843
+ 1205834(1 minus 119884
2)
4 1198894
1198891198844
(13)
where other derivatives can be derived in a similarmanner
Journal of Applied Mathematics 3
(v) The ansatz of the form
119880 (120583120585) = 119878 (119884) =
119872
sum
119896=0
119886119896119884119896+
119872
sum
119896=1
119887119896119884minus119896 (14)
is introduced where 119872 is a positive integer in mostcases that will be determined If119872 is not an integerthen a transformation formula is used to overcomethis difficulty Substituting (13) and (14) into the ODE(11) yields an equation in powers of 119884
(vi) To determine the parameter 119872 the linear terms ofthe highest order in the resulting equation with thehighest order nonlinear terms are balanced With119872determined one collects all coefficients of powers of119884 in the resulting equation where these coefficientshave to vanish This will give a system of algebraicequations involving the 119886
119896and 119887119896 (119896 = 0 119872) 119881
and 120583 Having determined these parameters knowingthat119872 is a positive integer in most cases and using(14) one obtains an analytic solution in a closed form
3 The Travelling Wave Solutions
31 The First Coupled We first consider the following twocoupled IBq equations
119906119905119905= (119906 + 2120572119906119908)
119909119909+
1
12
119906119909119909119905119905
119908119905119905= (2119908 + 120573119908
2+ 1205721199062)119909119909+
1
12
119908119909119909119905119905
(15)
Using the wave variable 120585 = 119909 minus 119881119905 then by integrating thisequation twice and considering the constants of integrationto be zero the system (15) is carried to a system of ODEs
12 (1198812minus 1)119880 minus 24120572119880119882 minus 119881
211988010158401015840= 0
12 (1198812minus 2)119882 minus 12120573119882
2minus 12120572119880
2minus 119881211988210158401015840= 0
(16)
Balancing 119881211988010158401015840 with 119880119882 and 119881211988210158401015840 with 1198802 in (16) gives
119872+ 2 = 119872 +119873
2119872 = 119873 + 2
(17)
so that119872 = 119873 = 2 We consider solutions in the form
119880 (120583120585) = 119878 (119884) =
2
sum
119896=0
119886119896119884119896+
2
sum
119896=1
119887119896119884minus119896
119882 (120583120585) = 119878 (119884) =
2
sum
119896=0
119888119896119884119896+
2
sum
119896=1
119889119896119884119896
(18)
Substituting (18) into the two components of (16) and col-lecting the coefficients of 119884 give two systems of algebraic
equations for 1198860 1198861 1198862 1198871 1198872 1198880 1198881 119888211988911198892119881 and120583 Solving
these systems together leads to the following sets
1198860= minus
1198862
3
1198862=
3radic120572120573 (1198814minus 31198812+ 2)
41205722
1198880=
120573 + 4120572
8120572 (120572 minus 120573)
1198882=
minus3120573
8120572 (120572 minus 120573)
119881 =
6
radicminus61205832+ 18
120583 =
radic3120573 (minus3120573 + 4120572)
minus3120573 + 4120572
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860= minus2119887
2 119886
2=
91205732(2120572 minus 120573)
102412057231198872(120572 minus 120573)
2
1198872=
3radic120572120573 (1198814minus 31198812+ 2)
161205722
1198880=
minus3120573 + 8120572
16120572 (120572 minus 120573)
1198882=
3120573
32120572 (120572 minus 120573)
1198892= 1198882
119881 =
6
radic241205832+ 18
120583 =
radic120573 (9120573 minus 12120572)
minus6120573 + 8120572
1198861= 1198871= 1198881= 1198891= 0
1198860=
21198872
3
1198862=
91205732(2120572 minus 120573)
102412057231198872(120572 minus 120573)
2
1198872=
3radic120572120573 (1198814minus 31198812+ 2)
161205722
1198880=
minus120573 + 8120572
16120572 (120572 minus 120573)
1198882=
minus3120573
32120572 (120572 minus 120573)
1198892= 1198882
119881 =
6
radicminus241205832+ 18
120583 =
radic3120573 (minus3120573 + 4120572)
minus6120573 + 8120572
1198861= 1198871= 1198881= 1198891= 0
1198860=
minus91205732(2120572 minus 120573)
6412057231198862(120572 minus 120573)
2 119886
2=
3radic120572120573 (1198814minus 31198812+ 2)
41205722
1198880=
minus3120573 + 4120572
8120572 (120572 minus 120573)
1198882=
3120573
8120572 (120572 minus 120573)
119881 =
6
radic61205832+ 18
120583 =
radic120573 (9120573 minus 12120572)
minus3120573 + 4120572
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
(19)
4 Journal of Applied Mathematics
Consequently we obtain the following travelling wave solu-tions
1199061(119909 119905) = minus
radic120572120573 (1198814minus 31198812+ 2)
41205722
+
3radic120572120573 (1198814minus 31198812+ 2)
41205722
tanh2radic3120573 (minus3120573 + 4120572)
minus3120573 + 4120572
times(119909 minus
6
radicminus61205832+ 18
119905)
1199081(119909 119905) =
120573 + 4120572
8120572 (120572 minus 120573)
minus
3120573
8120572 (120572 minus 120573)
tanh2radic3120573 (minus3120573 + 4120572)
minus3120573 + 4120572
times(119909 minus
6
radicminus61205832+ 18
119905)
1199062(119909 119905) = minus
3radic120572120573 (1198814minus 31198812+ 2)
81205722
+
91205732(2120572 minus 120573)
102412057231198872(120572 minus 120573)
2tanh2
radic120573 (9120573 minus 12120572)
minus6120573 + 8120572
times(119909 minus
6
radic241205832+ 18
119905)
+
3radic120572120573 (1198814minus 31198812+ 2)
161205722
coth2radic120573 (9120573 minus 12120572)
minus6120573 + 8120572
times(119909 minus
6
radic241205832+ 18
119905)
1199082(119909 119905) =
minus3120573 + 8120572
16120572 (120572 minus 120573)
+
3120573
32120572 (120572 minus 120573)
tanh2radic120573 (9120573 minus 12120572)
minus6120573 + 8120572
times(119909 minus
6
radic241205832+ 18
119905)
+
3120573
32120572 (120572 minus 120573)
coth2radic120573 (9120573 minus 12120572)
minus6120573 + 8120572
times(119909 minus
6
radic241205832+ 18
119905)
1199063(119909 119905) =
radic120572120573 (1198814minus 31198812+ 2)
81205722
+
91205732(2120572 minus 120573)
102412057231198872(120572 minus 120573)
2tanh2
radic3120573 (minus3120573 + 4120572)
minus6120573 + 8120572
times(119909 minus
6
radicminus241205832+ 18
119905)
+
3radic120572120573 (1198814minus 31198812+ 2)
161205722
coth2radic3120573 (minus3120573 + 4120572)
minus6120573 + 8120572
times(119909 minus
6
radicminus241205832+ 18
119905)
1199083(119909 119905) =
minus120573 + 8120572
16120572 (120572 minus 120573)
minus
3120573
32120572 (120572 minus 120573)
tanh2radic3120573 (minus3120573 + 4120572)
minus6120573 + 8120572
times(119909 minus
6
radicminus241205832+ 18
119905)
minus
3120573
32120572 (120572 minus 120573)
coth2radic3120573 (minus3120573 + 4120572)
minus6120573 + 8120572
times(119909 minus
6
radicminus241205832+ 18
119905)
1199064(119909 119905) =
minus91205732(2120572 minus 120573)
6412057231198862(120572 minus 120573)
2
+
3radic120572120573 (1198814minus 31198812+ 2)
41205722
tanh2radic120573 (9120573 minus 12120572)
minus3120573 + 4120572
times(119909 minus
6
radic61205832+ 18
119905)
Journal of Applied Mathematics 5
1199084(119909 119905) =
minus3120573 + 4120572
8120572 (120572 minus 120573)
+
3120573
8120572 (120572 minus 120573)
tanh2radic120573 (9120573 minus 12120572)
minus3120573 + 4120572
times(119909 minus
6
radic61205832+ 18
119905)
(20)
32 The Second Coupled We next consider the system ofequations
120588
120572
119906119905119905= 120573119906119909119909minus
1205732
2
(1199062)119909119909+
120573
2
(1199082)119909119909+
120588
120572
1198972
12
119906119909119909119905119905
120588
120572
119908119905119905= 120573(119906119908)
119909119909+
120588
120572
1198972
12
119908119909119909119905119905
(21)
Using thewave variable 120585 = 119909minus119881119905 and then by integrating thisequation twice and considering the constants of integration tobe zero the system (21) is carried to a system of ODEs
12 (120572120573 minus 1205881198812)119880 minus 6120572120573
21198802+ 6120572120573119882
2+ 1205881198972119881211988010158401015840= 0
12120572120573119880119882 + 1205881198972119881211988210158401015840minus 12120588119881
2119882 = 0
(22)
Balancing minus612057212057321198802 with 120588119897
2119881211988010158401015840 and 12120572120573119880119882 with
1205881198972119881211988210158401015840 in (22) gives
2119872 = 119872 + 2
2119873 = 119872 +119873
(23)
so that119872 = 119873 = 2 The tanh-coth method admits the use of
119880 (120583120585) = 119878 (119884) =
2
sum
119896=0
119886119896119884119896+
2
sum
119896=1
119887119896119884minus119896
119882 (120583120585) = 119878 (119884) =
2
sum
119896=0
119888119896119884119896+
2
sum
119896=1
119889119896119884119896
(24)
Substituting (24) into (22) collecting the coefficients of 119884and solving the resulting system we find the following setsof solutions
1198860=
1
120573 + 1
1198862=
minus1
2 (120573 + 1)
1198872= 1198862
1198892= plusmn
radic120573 + 2
2 (120573 + 1)
1198882= minus1198892
119881 =
radic3120588120572120573 (120573 + 1)
3120588 (120573 + 1)
120583 =
radic3
119897
1198861= 1198871= 1198880= 1198881= 1198891= 0
1198860=
1
120573 + 1
1198862=
3120573
61205732+ 14120573 + 8
1198872= 1198862
1198892= plusmn
3120573radic120573 + 2
2 (3120573 + 4) (120573 + 1)
1198882= minus1198892
119881 =
radic15120588120572120573 (211989721205832+ 9120573 + 15)
120588 (211989721205832+ 9120573 + 15)
120583 =
radicminus120573 (15120573 + 12)
(5120573 + 4) 119897
1198861= 1198871= 1198880= 1198881= 1198891= 0
1198860=
4 minus 120573
4 (120573 + 1)
1198862=
3120573
4120573 + 1
1198880=
minus1198882
3
1198882= plusmn
9120573
2radic611989721205832+ 36120573 + 18
119881 =
radicminus3120588120572120573 (minus3 + 11989721205832)
120588 (minus3 + 11989721205832)
120583 =
radicminus120573 (3120573 + 6)
(120573 + 2) 119897
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860=
3120573 + 4
4 (120573 + 1)
1198862=
minus3120573
4120573 + 1
1198880= minus1198882
1198882= plusmn
9120573
2radicminus611989721205832+ 36120573 + 18
119881 =
radic3120588120572120573 (3 + 11989721205832)
120588 (3 + 11989721205832)
120583 =
radic3120573 (120573 + 2)
(120573 + 2) 119897
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860=
120573 + 8
8 (120573 + 1)
1198862=
3120573
16 (120573 + 1)
1198872= 1198862 119888
0=
21198892
3
1198882= 1198892 119889
2= plusmn
9120573
8radic2411989721205832+ 36120573 + 18
119881 =
radicminus3120588120572120573 (minus3 + 411989721205832)
120588 (minus3 + 411989721205832)
120583 =
radicminus120573 (3120573 + 6)
2 (120573 + 2) 119897
1198861= 1198871= 1198881= 1198891= 0
1198860=
3 (120573 + 8)
8 (120573 + 1)
1198862=
minus3120573
16 (120573 + 1)
1198872= 1198862 119888
0= minus2119889
2
6 Journal of Applied Mathematics
1198882= 1198892 119889
2= plusmn
9120573
8radicminus2411989721205832+ 36120573 + 18
119881 =
radic3120588120572120573 (3 + 411989721205832)
120588 (3 + 411989721205832)
120583 =
radic120573 (3120573 + 6)
2 (120573 + 2) 119897
1198861= 1198871= 1198881= 1198891= 0
(25)
In accordance with these sets of solutions we have thefollowing solutions
1199061(119909 119905) =
1
120573 + 1
minus
1
2 (120573 + 1)
tanh2radic3
119897
times (119909 minus
radic3120588120572120573 (120573 + 1)
3120588 (120573 + 1)
119905)
minus
1
2 (120573 + 1)
coth2radic3
119897
(119909 minus
radic3120588120572120573 (120573 + 1)
3120588 (120573 + 1)
119905)
1199081(119909 119905) = ∓
radic120573 + 2
2 (120573 + 1)
tanh2radic3
119897
(119909 minus
radic3120588120572120573 (120573 + 1)
3120588 (120573 + 1)
119905)
plusmn
radic120573 + 2
2 (120573 + 1)
coth2radic3
119897
(119909 minus
radic3120588120572120573 (120573 + 1)
3120588 (120573 + 1)
119905)
1199062(119909 119905) =
1
120573 + 1
+
3120573
61205732+ 14120573 + 8
tanh2radicminus120573 (15120573 + 12)
(5120573 + 4) 119897
times (119909 minus
radic15120588120572120573 (211989721205832+ 9120573 + 15)
120588 (211989721205832+ 9120573 + 15)
119905)
+
3120573
61205732+ 14120573 + 8
coth2radicminus120573 (15120573 + 12)
(5120573 + 4) 119897
times (119909 minus
radic15120588120572120573 (211989721205832+ 9120573 + 15)
120588 (211989721205832+ 9120573 + 15)
119905)
1199082(119909 119905) = ∓
3120573radic120573 + 2
2 (3120573 + 4) (120573 + 1)
tanh2radicminus120573 (15120573 + 12)
(5120573 + 4) 119897
times (119909 minus
radic15120588120572120573 (211989721205832+ 9120573 + 15)
120588 (211989721205832+ 9120573 + 15)
119905)
plusmn
3120573radic120573 + 2
2 (3120573 + 4) (120573 + 1)
coth2radicminus120573 (15120573 + 12)
(5120573 + 4) 119897
times (119909 minus
radic15120588120572120573 (211989721205832+ 9120573 + 15)
120588 (211989721205832+ 9120573 + 15)
119905)
1199063(119909 119905) =
4 minus 120573
4 (120573 + 1)
+
3120573
4120573 + 1
tanh2radicminus120573 (3120573 + 6)
(120573 + 2) 119897
times (119909 minus
radicminus3120588120572120573 (minus3 + 11989721205832)
120588 (minus3 + 11989721205832)
119905)
1199083(119909 119905) = ∓
3120573
2radic611989721205832+ 36120573 + 18
plusmn
9120573
2radic611989721205832+ 36120573 + 18
tanh2radicminus120573 (3120573 + 6)
(120573 + 2) 119897
times (119909 minus
radicminus3120588120572120573 (minus3 + 11989721205832)
120588 (minus3 + 11989721205832)
119905)
1199064(119909 119905) =
3120573 + 4
4 (120573 + 1)
minus
3120573
4120573 + 1
tanh2radic3120573 (120573 + 2)
(120573 + 2) 119897
times (119909 minus
radic3120588120572120573 (3 + 11989721205832)
120588 (3 + 11989721205832)
119905)
1199084(119909 119905) = ∓
9120573
2radicminus611989721205832+ 36120573 + 18
plusmn
9120573
2radicminus611989721205832+ 36120573 + 18
tanh2radic3120573 (120573 + 2)
(120573 + 2) 119897
times (119909 minus
radic3120588120572120573 (3 + 11989721205832)
120588 (3 + 11989721205832)
119905)
1199065(119909 119905) =
120573 + 8
8 (120573 + 1)
+
3120573
16 (120573 + 1)
tanh2radicminus120573 (3120573 + 6)
2 (120573 + 2) 119897
Journal of Applied Mathematics 7
times(119909 minus
radicminus3120588120572120573 (minus3 + 411989721205832)
120588 (minus3 + 411989721205832)
119905)
+
3120573
16 (120573 + 1)
coth2radicminus120573 (3120573 + 6)
2 (120573 + 2) 119897
times (119909 minus
radicminus3120588120572120573 (minus3 + 411989721205832)
120588 (minus3 + 411989721205832)
119905)
1199085(119909 119905) = plusmn
3120573
4radic2411989721205832+ 36120573 + 18
plusmn
9120573
8radic2411989721205832+ 36120573 + 18
tanh2radicminus120573 (3120573 + 6)
2 (120573 + 2) 119897
times (119909 minus
radicminus3120588120572120573 (minus3 + 411989721205832)
120588 (minus3 + 411989721205832)
119905)
plusmn
9120573
8radic2411989721205832+ 36120573 + 18
coth2radicminus120573 (3120573 + 6)
2 (120573 + 2) 119897
times (119909 minus
radicminus3120588120572120573 (minus3 + 411989721205832)
120588 (minus3 + 411989721205832)
119905)
1199066(119909 119905) =
3 (120573 + 8)
8 (120573 + 1)
minus
3120573
16 (120573 + 1)
tanh2radic120573 (3120573 + 6)
2 (120573 + 2) 119897
times (119909 minus
radic3120588120572120573 (3 + 411989721205832)
120588 (3 + 411989721205832)
119905)
minus
3120573
16 (120573 + 1)
coth2radic120573 (3120573 + 6)
2 (120573 + 2) 119897
times (119909 minus
radic3120588120572120573 (3 + 411989721205832)
120588 (3 + 411989721205832)
119905)
1199086(119909 119905) = ∓
9120573
4radicminus2411989721205832+ 36120573 + 18
plusmn
9120573
8radicminus2411989721205832+ 36120573 + 18
tanh2radic120573 (3120573 + 6)
2 (120573 + 2) 119897
times (119909 minus
radic3120588120572120573 (3 + 411989721205832)
120588 (3 + 411989721205832)
119905)
plusmn
9120573
8radicminus2411989721205832+ 36120573 + 18
coth2radic120573 (3120573 + 6)
2 (120573 + 2) 119897
times (119909 minus
radic3120588120572120573 (3 + 411989721205832)
120588 (3 + 411989721205832)
119905)
(26)
33 The Third Coupled We now examine the coupled equa-tions
119906119905119905= 120573119906119909119909minus
120573
2
(1199062)119909119909+
1
2
(1199082)119909119909+ 1205722119906119909119909119905119905
119908119905119905= (119906119908)
119909119909+ 1205722119908119909119909119905119905
(27)
Using the wave variable 120585 = 119909 minus119881119905 the system (27) is carriedto a system of ODEs Integrating these equations twice andneglecting constants of integration we find
2 (1198812minus 120573)119880 + 120573119880
2minus 120573119882
2minus 21205722119881211988010158401015840= 0
1198812119882minus119880119882 minus 120572
2119881211988210158401015840= 0
(28)
Balancing 1205731198822 with 21205722119881211988010158401015840 and 119881211988210158401015840 with 119880119882 in (28)gives
2119873 = 119872 + 2
2 + 119873 = 119872 +119873
(29)
so that
119872 = 119873 = 2 (30)
As a result we seek solutions to (27) in the form
119880(120583120585) = 119878 (119884) =
2
sum
119896=0
119886119896119884119896+
2
sum
119896=1
119887119896119884minus119896
119882 (120583120585) = 119878 (119884) =
2
sum
119896=0
119888119896119884119896+
2
sum
119896=1
119889119896119884119896
(31)
Substitution of (31) into (28) leads to an algebraic system inthe unknowns 119886
0 1198861 1198862 1198871 1198872 1198880 1198881 1198882 1198891 1198892 119881 and 120583
8 Journal of Applied Mathematics
Collecting the coefficients of 119884 and solving the resultingsystem we find the following sets of solutions
1198860=
120573 (3120573 + 8)
8 (120573 + 1)
1198862=
minus31205732
16 (120573 + 1)
1198872= 1198862 119888
0= minus2119889
2 119888
2= 1198892
1198892=
plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573
16
119881 =
radic120573 (1612057221205832+ 1)
1612057221205832+ 1
120583 =
radic120573 (120573 + 2)
4120572 (120573 + 2)
1198861= 1198871= 1198881= 1198891= 0
1198860=
120573 (120573 + 8)
8 (120573 + 1)
1198862=
31205732
16 (120573 + 1)
1198872= 1198862
1198880=
21198892
3
1198882= 1198892
1198892=
plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573
16
119881 =
radicminus120573 (1612057221205832minus 1)
1612057221205832minus 1
120583 =
radicminus120573 (120573 + 2)
4120572 (120573 + 2)
1198861= 1198871= 1198881= 1198891= 0
1198860=
minus120573 (120573 minus 4)
4 (120573 + 1)
1198862=
31205732
4 (120573 + 1)
1198880= minus
1198882
3
1198882=
plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573
4
119881 =
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
120583 =
radicminus120573 (120573 + 2)
2120572 (120573 + 2)
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860=
120573 (3120573 + 4)
4 (120573 + 1)
1198862=
minus31205732
4 (120573 + 1)
1198880= minus2
1198882=
plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573
4
119881 =
radic120573 (412057221205832+ 1)
412057221205832+ 1
120583 =
radic120573 (120573 + 2)
2120572 (120573 + 2)
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860=
120573
120573 + 1
1198862=
31205732
2 (31205732+ 7120573 + 4)
1198872= 1198862 119888
2= minus1198892
1198892=
plusmn3radicminus421198814120573 minus 58119881
4minus 21198812120573 minus 120119881
2+ 120120573
8
119881 =
radic5120573 (812057221205832+ 3120573 + 5)
812057221205832+ 3120573 + 5
120583 =
radicminus120573 (5120573 + 4)
2120572 (5120573 + 4)
1198861= 1198871= 1198880= 1198881= 1198891= 0
1198860=
120573
120573 + 1
1198862=
minus120573
2 (120573 + 1)
1198872= 1198862
1198882= minus1198892 119889
2=
plusmnradic91198812+ 6
2
119881 =
radic3120573 (120573 + 1)
3 (120573 + 1)
120583 =
minus1
2120572
1198861= 1198871= 1198880= 1198881= 1198891= 0
(32)
Consequently we obtain the following solutions
1199061(119909 119905) =
120573 (3120573 + 8)
8 (120573 + 1)
minus
31205732
16 (120573 + 1)
tanh2radic120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radic120573 (1612057221205832+ 1)
1612057221205832+ 1
119905)
minus
31205732
16 (120573 + 1)
coth2radic120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radic120573 (1612057221205832+ 1)
1612057221205832+ 1
119905)
1199081(119909 119905) =
∓3radicminus41198814minus 41198812+ 21205732+ 4120573
8
plusmn
3radicminus41198814minus 41198812+ 21205732+ 4120573
16
tanh2radic120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radic120573 (1612057221205832+ 1)
1612057221205832+ 1
119905)
plusmn
3radicminus41198814minus 41198812+ 21205732+ 4120573
16
coth2radic120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radic120573 (1612057221205832+ 1)
1612057221205832+ 1
119905)
Journal of Applied Mathematics 9
1199062(119909 119905) =
120573 (120573 + 8)
8 (120573 + 1)
+
31205732
16 (120573 + 1)
tanh2radicminus120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radicminus120573 (1612057221205832minus 1)
1612057221205832minus 1
119905)
+
31205732
16 (120573 + 1)
coth2radicminus120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radicminus120573 (1612057221205832minus 1)
1612057221205832minus 1
119905)
1199082(119909 119905) = plusmn
radicminus41198814minus 41198812+ 21205732+ 4120573
8
plusmn
3radicminus41198814minus 41198812+ 21205732+ 4120573
16
tanh2radicminus120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radicminus120573 (1612057221205832minus 1)
1612057221205832minus 1
119905)
plusmn
3radicminus41198814minus 41198812+ 21205732+ 4120573
16
coth2radicminus120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radicminus120573 (1612057221205832minus 1)
1612057221205832minus 1
119905)
1199063(119909 119905) =
minus120573 (120573 minus 4)
4 (120573 + 1)
+
31205732
4 (120573 + 1)
tanh2radicminus120573 (120573 + 2)
2120572 (120573 + 2)
times (119909 minus
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
119905)
1199083(119909 119905) = ∓
radicminus41198814minus 41198812+ 21205732+ 4120573
4
plusmn
3radicminus41198814minus 41198812+ 21205732+ 4120573
4
tanhradicminus120573 (120573 + 2)
2120572 (120573 + 2)
times (119909 minus
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
119905)
1199064(119909 119905) =
120573 (3120573 + 4)
4 (120573 + 1)
minus
31205732
4 (120573 + 1)
tanh2radic120573 (120573 + 2)
2120572 (120573 + 2)
times (119909 minus
radic120573 (412057221205832+ 1)
412057221205832+ 1
119905)
1199084(119909 119905) = ∓
3radicminus41198814minus 41198812+ 21205732+ 4120573
4
plusmn
3radicminus41198814minus 41198812+ 21205732+ 4120573
4
tanh2radic120573 (120573 + 2)
2120572 (120573 + 2)
times (119909 minus
radic120573 (412057221205832+ 1)
412057221205832+ 1
119905)
1199065(119909 119905) =
120573
120573 + 1
+
31205732
2 (31205732+ 7120573 + 4)
tanh2radicminus120573 (5120573 + 4)
2120572 (5120573 + 4)
times (119909 minus
radic5120573 (812057221205832+ 3120573 + 5)
812057221205832+ 3120573 + 5
119905)
+
31205732
2 (31205732+ 7120573 + 4)
coth2radicminus120573 (5120573 + 4)
2120572 (5120573 + 4)
times (119909 minus
radic5120573 (812057221205832+ 3120573 + 5)
812057221205832+ 3120573 + 5
119905)
1199085(119909 119905)
= ∓
3radicminus421198814120573 minus 58119881
4minus 21198812120573 minus 120119881
2+ 120120573
8
times tanh2radicminus120573 (5120573 + 4)
2120572 (5120573 + 4)
times (119909 minus
radic5120573 (812057221205832+ 3120573 + 5)
812057221205832+ 3120573 + 5
119905)
plusmn
3radicminus421198814120573 minus 58119881
4minus 21198812120573 minus 120119881
2+ 120120573
8
10 Journal of Applied Mathematics
times coth2radicminus120573 (5120573 + 4)
2120572 (5120573 + 4)
times (119909 minus
radic5120573 (812057221205832+ 3120573 + 5)
812057221205832+ 3120573 + 5
119905)
1199066(119909 119905) =
120573
120573 + 1
minus
120573
2 (120573 + 1)
tanh2minus12120572
(119909 minus
radic3120573 (120573 + 1)
3 (120573 + 1)
119905)
minus
120573
2 (120573 + 1)
coth2minus12120572
(119909 minus
radic3120573 (120573 + 1)
3 (120573 + 1)
119905)
1199086(119909 119905)
= ∓
radic91198812+ 6
2
tanh2minus12120572
(119909 minus
radic3120573 (120573 + 1)
3 (120573 + 1)
119905)
plusmn
radic91198812+ 6
2
coth2minus12120572
(119909 minus
radic3120573 (120573 + 1)
3 (120573 + 1)
119905)
(33)
34 The Fourth Coupled We now consider the coupled IBqequations
119906119905119905= 119906119909119909minus 120573(119906
2)119909119909+ (1199082)119909119909+ 119906119909119909119905119905
119908119905119905= 119908119909119909+ 2(119906119908)
119909119909+ 119908119909119909119905119905
(34)
Using the wave variable 120585 = 119909 minus 119881119905 (34) can be converted toa system of ODEs
(1198812minus 1)119880 + 120573119880
2minus1198822minus 119881211988010158401015840= 0
(1198812minus 1)119882 minus 2119880119882 minus 119881
211988210158401015840= 0
(35)
Balancing1198822 with 11988010158401015840 and 119880119882 with11988210158401015840 in (35) gives
2119873 = 119872 + 2
119872 + 119873 = 119873 + 2
(36)
so that
119872 = 119873 = 2 (37)
The tanh-coth method substitutes the finite expansions
119880(120583120585) = 119878 (119884) =
2
sum
119896=0
119886119896119884119896+
2
sum
119896=1
119887119896119884minus119896
119882 (120583120585) = 119878 (119884) =
2
sum
119896=0
119888119896119884119896+
2
sum
119896=1
119889119896119884119896
(38)
In (35) collecting the coefficients of each power of 119884 andsolving the resulting system we obtain
1198860=
minus21205832
161205832+ 1
1198862=
minus31205832
161205832+ 1
1198862= 1198872
1198880=
21198892
3
1198882= 1198892
1198892=
plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
119881 =
1
radic161205832+ 1
1198861= 1198871= 1198881= 1198891= 0
1198860=
minus61205832
161205832minus 1
1198862=
31205832
161205832minus 1
1198862= 1198872
1198880= minus2119889
2 119888
2= 1198892
1198892=
plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
119881 =
1
radicminus161205832+ 1
1198861= 1198871= 1198881= 1198891= 0
1198860=
1205832
41205832+ 1
1198862=
minus31205832
41205832+ 1
1198880=
minus1198882
3
1198882=
plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
119881 =
1
radic41205832+ 1
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860=
minus31205832
41205832minus 1
1198862=
31205832
41205832minus 1
1198880= minus1198882
1198882=
plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
119881 =
1
radicminus41205832+ 1
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860=
minus31205832
41205832minus 1
1198872=
31205832
41205832minus 1
1198880= minus1198892
1198892=
plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
119881 =
1
radicminus41205832+ 1
1198861= 1198862= 1198871= 1198881= 1198882= 1198891= 0
(39)
Journal of Applied Mathematics 11
where 120583 is left as a free parameter Consequently we obtainthe following travelling wave solutions
1199061(119909 119905) = minus
21205832
161205832+ 1
minus
31205832
161205832+ 1
tanh2120583(119909 minus 1
radic161205832+ 1
119905)
minus
31205832
161205832+ 1
coth2120583(119909 minus 1
radic161205832+ 1
119905)
1199081(119909 119905) = plusmn
(119881 + 1) (119881 minus 1)radic120573 + 2
8
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
tanh2120583
times(119909 minus
1
radic161205832+ 1
119905)
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
coth2120583
times(119909 minus
1
radic161205832+ 1
119905)
1199062(119909 119905) = minus
61205832
161205832minus 1
+
31205832
161205832minus 1
tanh2120583(119909 minus 1
radicminus161205832+ 1
119905)
+
31205832
161205832minus 1
coth2120583(119909 minus 1
radicminus161205832+ 1
119905)
1199082(119909 119905) = ∓
3 (119881 + 1) (119881 minus 1)radic120573 + 2
8
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
tanh2120583
times(119909 minus
1
radicminus161205832+ 1
119905)
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
coth2120583
times(119909 minus
1
radicminus161205832+ 1
119905)
1199063(119909 119905) =
1205832
41205832+ 1
minus
31205832
41205832+ 1
tanh2120583(119909 minus 1
radic41205832+ 1
119905)
1199083(119909 119905) = ∓
(119881 + 1) (119881 minus 1)radic120573 + 2
4
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
times tanh2120583(119909 minus 1
radic41205832+ 1
119905)
1199064(119909 119905) = minus
31205832
41205832minus 1
+
31205832
41205832minus 1
tanh2120583(119909 minus 1
radicminus41205832+ 1
119905)
1199084(119909 119905) = ∓
3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
times tanh2120583(119909 minus 1
radicminus41205832+ 1
119905)
1199065(119909 119905) = minus
31205832
41205832minus 1
+
31205832
41205832minus 1
coth2120583(119909 minus 1
radicminus41205832+ 1
119905)
1199085(119909 119905) = ∓
3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
plusmn 3
(119881 + 1) (119881 minus 1)radic120573 + 2
4
times coth2120583(119909 minus 1
radicminus41205832+ 1
119905)
(40)
35 The Fifth Coupled We finally consider the coupled IBqequations
119906119905119905= 120573119906119909119909minus
120573
2
(1199062)119909119909+
120573
2
(1199082)119909119909+ 1205722119906119909119909119905119905
119908119905119905= 120574(119906119908)
119909119909+ 120588(119908
2)119909119909+ 1205722119908119909119909119905119905
(41)
12 Journal of Applied Mathematics
This system of equations (41) can be converted to the follow-ing system of ODEs by using the wave variable 120585 = 119909 minus 119881119905
(21198812minus 2120573)119880 + 120573119880
2minus 120573119882
2minus 21205722119881211988010158401015840= 0
1198812119882minus 120574119880119882 minus 120588119882
2minus 1205722119881211988210158401015840= 0
(42)
Using the balancing procedure in (42) gives
2119872 = 119872 +119873
2119873 = 119872 +119873
(43)
so that
119872 = 119873 = 2 (44)
We consider solutions in the following form
119880 (120583120585) = 119878 (119884) =
2
sum
119896=0
119886119896119884119896+
2
sum
119896=1
119887119896119884minus119896
119882 (120583120585) = 119878 (119884) =
2
sum
119896=0
119888119896119884119896+
2
sum
119896=1
119889119896119884119896
(45)
Substituting (45) into (42) collecting the coefficients of 119884and solving the resulting system we find the following setsof solutions
1198860= (minus120573 (8120588119888
21205732120574 minus 8120573120574
21205881198882minus 18120588
21205732+ 24120573120588
31198882
+1212057321205742minus 16120574
31205881198882+ 24120588
31198882120574 minus 3120574120573
3))
times (4 (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882+ 312058821205733
+ 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882minus 312057331205742minus 312057431205732))
minus1
1198862=
minus (41205881198882120573 + 4119888
2120588120574 minus 3120573
2)
4120574 (120583 + 120573)
1198880= (120573(120588
212057321198882+ 120574211988821205732+ 12057311988821205743+ 2120574212058821198882
+312057412058821205731198882minus
3
2
1205881205732120574 minus
3
4
1205881205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882+ 312058821205733
+ 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882minus 312057331205742minus 312057431205732)
minus1
1198882= (plusmn3 (minus 120573120588 + (minus120588
21205732+ 41205743120573 minus 4120574
21198814
minus 412057431198812+ 212057321205742
minus41205882120574120573 + 4120588
21198814+ 412058821198812120574)
12
))
times (4 (1205742minus 1205882))
minus1
119881 =
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
120583 =
radicminus120573 (2120574 + 120573)
2120572 (2120574 + 120573)
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860= (120573(6120588119888
21205732120574 + 10120573120574
21205881198882+
3
2
12058821205732+ 2120573120588
31198882
+312057321205742+ 412057431205881198882+ 212058831198882120574 +
9
4
1205741205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882minus 312058821205733
minus 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882+ 312057331205742+ 312057431205732)
minus1
1198862=
minus (41205881198882120573 + 4119888
2120588120574 + 3120573
2)
4120574 (120583 + 120573)
1198880= (120573(minus 3120588
212057321198882minus 3120574211988821205732minus 312057311988821205743
minus6120574212058821198882minus 9120574120588
21205731198882minus
9
2
1205881205732120574 minus
9
4
1205881205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882
minus 312058821205733minus 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882+ 312057331205742+ 312057431205732)
minus1
1198882= (plusmn3 (120573120588 + (minus120588
21205732+ 41205743120573 minus 4120574
21198814
minus 412057431198812+ 212057321205742minus 41205882120574120573
+412058821198814+ 412058821198812120574)
12
))
times (4 (1205742minus 1205882))
minus1
119881 =
radic120573 (412057221205832+ 1)
412057221205832+ 1
120583 =
radic120573 (2120574 + 120573)
2120572 (2120574 + 120573)
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
(46)
Journal of Applied Mathematics 13
In view of this we obtain the following travelling wavesolutions
1199061(119909 119905) = (minus120573 (8120588119888
21205732120574 minus 8120573120574
21205881198882minus 18120588
21205732+ 24120573120588
31198882
+1212057321205742minus 16120574
31205881198882+ 24120588
31198882120574 minus 3120574120573
3))
times (4 (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882+ 312058821205733
+ 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882minus 312057331205742minus 312057431205732))
minus1
minus
(41205881198882120573 + 4119888
2120588120574 minus 3120573
2)
4120574 (120583 + 120573)
tanh2radicminus120573 (2120574 + 120573)
2120572 (2120574 + 120573)
times(119909 minus
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
119905)
1199081(119909 119905) = (120573 (120588
212057321198882+ 120574211988821205732+ 12057311988821205743+ 2120574212058821198882
+312057412058821205731198882minus
3
2
1205881205732120574 minus
3
4
1205881205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882+ 312058821205733
+ 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882minus 312057331205742minus 312057431205732)
minus1
plusmn (3 (minus 120573120588 + (minus12058821205732+ 41205743120573 minus 4120574
21198814
minus 412057431198812+ 212057321205742minus 41205882120574120573
+412058821198814+ 412058821198812120574)
12
))
times (4 (1205742minus 1205882))
minus1
times tanh2radicminus120573 (2120574 + 120573)
2120572 (2120574 + 120573)
times(119909 minus
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
119905)
1199062(119909 119905) = (120573 (6120588119888
21205732120574 + 10120573120574
21205881198882+
3
2
12058821205732+ 2120573120588
31198882
+312057321205742+ 412057431205881198882+ 212058831198882120574 +
9
4
1205741205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882minus 312058821205733
minus 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882+ 312057331205742+ 312057431205732)
minus1
minus
(41205881198882120573 + 4119888
2120588120574 + 3120573
2)
4120574 (120583 + 120573)
tanh2radic120573 (2120574 + 120573)
2120572 (2120574 + 120573)
times(119909 minus
radic120573 (412057221205832+ 1)
412057221205832+ 1
119905)
1199082(119909 119905) = (120573 (minus 3120588
212057321198882minus 3120574211988821205732minus 312057311988821205743
minus6120574212058821198882minus 9120574120588
21205731198882minus
9
2
1205881205732120574 minus
9
4
1205881205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882minus 312058821205733
minus 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882+ 312057331205742+ 312057431205732)
minus1
plusmn (3 (120573120588 + (minus12058821205732+ 41205743120573 minus 4120574
21198814
minus 412057431198812+ 212057321205742minus 41205882120574120573
+412058821198814+ 412058821198812120574)
12
))
times (4 (1205742minus 1205882))
minus1
times tanh2radic120573 (2120574 + 120573)
2120572 (2120574 + 120573)
(119909 minus
radic120573 (412057221205832+ 1)
412057221205832+ 1
119905)
(47)
4 Conclusion
In summary the tanh-coth method has been successfullyimplemented to find new travelling wave solutions for severalkinds of two coupled improved Boussinesq equations Thesolutions that have been found can be seen as an expansionand verification of the previously known data However thisstudy confirmed again the belief that the tanh-coth methodis a powerful and reliable technique to handle nonlineardispersive-dissipative equations and system of equationsThroughout the work Maple has been used to overcome thetedious algebraic calculations
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] A De Godefroy ldquoBlow up of solutions of a generalized Boussi-nesq equationrdquo IMA Journal of AppliedMathematics vol 60 no2 pp 123ndash138 1998
14 Journal of Applied Mathematics
[2] S Wang and M Li ldquoThe Cauchy problem for coupled IMBqequationsrdquo IMA Journal of Applied Mathematics vol 74 no 5pp 726ndash740 2009
[3] N Duruk H A Erbay and A Erkip ldquoBlow-up and globalexistence for a general class of nonlocal nonlinear coupled waveequationsrdquo Journal of Differential Equations vol 250 no 3 pp1448ndash1459 2011
[4] G Chen and H Zhang ldquoInitial boundary value problem for asystem of generalized IMBq equationsrdquoMathematical Methodsin the Applied Sciences vol 27 no 5 pp 497ndash518 2004
[5] P Rosenau ldquoDynamics of dense latticesrdquo Physical Review B vol36 no 11 pp 5868ndash5876 1987
[6] P L Christiansen P S Lomdahl andVMuto ldquoOn aToda latticemodel with a transversal degree of freedomrdquo Nonlinearity vol4 no 2 pp 477ndash501 1991
[7] S K Turitsyn ldquoOn a Toda lattice model with a transversaldegree of freedom Sufficient criterion of blow-up in thecontinuum limitrdquo Physics Letters A vol 173 no 3 pp 267ndash2691993
[8] R L Pego P Smereka and M I Weinstein ldquoOscillatoryinstability of solitary waves in a continuum model of latticevibrationsrdquo Nonlinearity vol 8 no 6 pp 921ndash941 1995
[9] X Fan and L Tian ldquoCompacton solutions and multiple com-pacton solutions for a continuum Toda lattice modelrdquo ChaosSolitons and Fractals vol 29 no 4 pp 882ndash894 2006
[10] A M Wazwaz ldquoThe Hirotarsquos bilinear method and the tanh-coth method for multiple-soliton solutions of the Sawada-Kotera-Kadomtsev-Petviashvili equationrdquoAppliedMathematicsand Computation vol 200 no 1 pp 160ndash166 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 3
(v) The ansatz of the form
119880 (120583120585) = 119878 (119884) =
119872
sum
119896=0
119886119896119884119896+
119872
sum
119896=1
119887119896119884minus119896 (14)
is introduced where 119872 is a positive integer in mostcases that will be determined If119872 is not an integerthen a transformation formula is used to overcomethis difficulty Substituting (13) and (14) into the ODE(11) yields an equation in powers of 119884
(vi) To determine the parameter 119872 the linear terms ofthe highest order in the resulting equation with thehighest order nonlinear terms are balanced With119872determined one collects all coefficients of powers of119884 in the resulting equation where these coefficientshave to vanish This will give a system of algebraicequations involving the 119886
119896and 119887119896 (119896 = 0 119872) 119881
and 120583 Having determined these parameters knowingthat119872 is a positive integer in most cases and using(14) one obtains an analytic solution in a closed form
3 The Travelling Wave Solutions
31 The First Coupled We first consider the following twocoupled IBq equations
119906119905119905= (119906 + 2120572119906119908)
119909119909+
1
12
119906119909119909119905119905
119908119905119905= (2119908 + 120573119908
2+ 1205721199062)119909119909+
1
12
119908119909119909119905119905
(15)
Using the wave variable 120585 = 119909 minus 119881119905 then by integrating thisequation twice and considering the constants of integrationto be zero the system (15) is carried to a system of ODEs
12 (1198812minus 1)119880 minus 24120572119880119882 minus 119881
211988010158401015840= 0
12 (1198812minus 2)119882 minus 12120573119882
2minus 12120572119880
2minus 119881211988210158401015840= 0
(16)
Balancing 119881211988010158401015840 with 119880119882 and 119881211988210158401015840 with 1198802 in (16) gives
119872+ 2 = 119872 +119873
2119872 = 119873 + 2
(17)
so that119872 = 119873 = 2 We consider solutions in the form
119880 (120583120585) = 119878 (119884) =
2
sum
119896=0
119886119896119884119896+
2
sum
119896=1
119887119896119884minus119896
119882 (120583120585) = 119878 (119884) =
2
sum
119896=0
119888119896119884119896+
2
sum
119896=1
119889119896119884119896
(18)
Substituting (18) into the two components of (16) and col-lecting the coefficients of 119884 give two systems of algebraic
equations for 1198860 1198861 1198862 1198871 1198872 1198880 1198881 119888211988911198892119881 and120583 Solving
these systems together leads to the following sets
1198860= minus
1198862
3
1198862=
3radic120572120573 (1198814minus 31198812+ 2)
41205722
1198880=
120573 + 4120572
8120572 (120572 minus 120573)
1198882=
minus3120573
8120572 (120572 minus 120573)
119881 =
6
radicminus61205832+ 18
120583 =
radic3120573 (minus3120573 + 4120572)
minus3120573 + 4120572
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860= minus2119887
2 119886
2=
91205732(2120572 minus 120573)
102412057231198872(120572 minus 120573)
2
1198872=
3radic120572120573 (1198814minus 31198812+ 2)
161205722
1198880=
minus3120573 + 8120572
16120572 (120572 minus 120573)
1198882=
3120573
32120572 (120572 minus 120573)
1198892= 1198882
119881 =
6
radic241205832+ 18
120583 =
radic120573 (9120573 minus 12120572)
minus6120573 + 8120572
1198861= 1198871= 1198881= 1198891= 0
1198860=
21198872
3
1198862=
91205732(2120572 minus 120573)
102412057231198872(120572 minus 120573)
2
1198872=
3radic120572120573 (1198814minus 31198812+ 2)
161205722
1198880=
minus120573 + 8120572
16120572 (120572 minus 120573)
1198882=
minus3120573
32120572 (120572 minus 120573)
1198892= 1198882
119881 =
6
radicminus241205832+ 18
120583 =
radic3120573 (minus3120573 + 4120572)
minus6120573 + 8120572
1198861= 1198871= 1198881= 1198891= 0
1198860=
minus91205732(2120572 minus 120573)
6412057231198862(120572 minus 120573)
2 119886
2=
3radic120572120573 (1198814minus 31198812+ 2)
41205722
1198880=
minus3120573 + 4120572
8120572 (120572 minus 120573)
1198882=
3120573
8120572 (120572 minus 120573)
119881 =
6
radic61205832+ 18
120583 =
radic120573 (9120573 minus 12120572)
minus3120573 + 4120572
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
(19)
4 Journal of Applied Mathematics
Consequently we obtain the following travelling wave solu-tions
1199061(119909 119905) = minus
radic120572120573 (1198814minus 31198812+ 2)
41205722
+
3radic120572120573 (1198814minus 31198812+ 2)
41205722
tanh2radic3120573 (minus3120573 + 4120572)
minus3120573 + 4120572
times(119909 minus
6
radicminus61205832+ 18
119905)
1199081(119909 119905) =
120573 + 4120572
8120572 (120572 minus 120573)
minus
3120573
8120572 (120572 minus 120573)
tanh2radic3120573 (minus3120573 + 4120572)
minus3120573 + 4120572
times(119909 minus
6
radicminus61205832+ 18
119905)
1199062(119909 119905) = minus
3radic120572120573 (1198814minus 31198812+ 2)
81205722
+
91205732(2120572 minus 120573)
102412057231198872(120572 minus 120573)
2tanh2
radic120573 (9120573 minus 12120572)
minus6120573 + 8120572
times(119909 minus
6
radic241205832+ 18
119905)
+
3radic120572120573 (1198814minus 31198812+ 2)
161205722
coth2radic120573 (9120573 minus 12120572)
minus6120573 + 8120572
times(119909 minus
6
radic241205832+ 18
119905)
1199082(119909 119905) =
minus3120573 + 8120572
16120572 (120572 minus 120573)
+
3120573
32120572 (120572 minus 120573)
tanh2radic120573 (9120573 minus 12120572)
minus6120573 + 8120572
times(119909 minus
6
radic241205832+ 18
119905)
+
3120573
32120572 (120572 minus 120573)
coth2radic120573 (9120573 minus 12120572)
minus6120573 + 8120572
times(119909 minus
6
radic241205832+ 18
119905)
1199063(119909 119905) =
radic120572120573 (1198814minus 31198812+ 2)
81205722
+
91205732(2120572 minus 120573)
102412057231198872(120572 minus 120573)
2tanh2
radic3120573 (minus3120573 + 4120572)
minus6120573 + 8120572
times(119909 minus
6
radicminus241205832+ 18
119905)
+
3radic120572120573 (1198814minus 31198812+ 2)
161205722
coth2radic3120573 (minus3120573 + 4120572)
minus6120573 + 8120572
times(119909 minus
6
radicminus241205832+ 18
119905)
1199083(119909 119905) =
minus120573 + 8120572
16120572 (120572 minus 120573)
minus
3120573
32120572 (120572 minus 120573)
tanh2radic3120573 (minus3120573 + 4120572)
minus6120573 + 8120572
times(119909 minus
6
radicminus241205832+ 18
119905)
minus
3120573
32120572 (120572 minus 120573)
coth2radic3120573 (minus3120573 + 4120572)
minus6120573 + 8120572
times(119909 minus
6
radicminus241205832+ 18
119905)
1199064(119909 119905) =
minus91205732(2120572 minus 120573)
6412057231198862(120572 minus 120573)
2
+
3radic120572120573 (1198814minus 31198812+ 2)
41205722
tanh2radic120573 (9120573 minus 12120572)
minus3120573 + 4120572
times(119909 minus
6
radic61205832+ 18
119905)
Journal of Applied Mathematics 5
1199084(119909 119905) =
minus3120573 + 4120572
8120572 (120572 minus 120573)
+
3120573
8120572 (120572 minus 120573)
tanh2radic120573 (9120573 minus 12120572)
minus3120573 + 4120572
times(119909 minus
6
radic61205832+ 18
119905)
(20)
32 The Second Coupled We next consider the system ofequations
120588
120572
119906119905119905= 120573119906119909119909minus
1205732
2
(1199062)119909119909+
120573
2
(1199082)119909119909+
120588
120572
1198972
12
119906119909119909119905119905
120588
120572
119908119905119905= 120573(119906119908)
119909119909+
120588
120572
1198972
12
119908119909119909119905119905
(21)
Using thewave variable 120585 = 119909minus119881119905 and then by integrating thisequation twice and considering the constants of integration tobe zero the system (21) is carried to a system of ODEs
12 (120572120573 minus 1205881198812)119880 minus 6120572120573
21198802+ 6120572120573119882
2+ 1205881198972119881211988010158401015840= 0
12120572120573119880119882 + 1205881198972119881211988210158401015840minus 12120588119881
2119882 = 0
(22)
Balancing minus612057212057321198802 with 120588119897
2119881211988010158401015840 and 12120572120573119880119882 with
1205881198972119881211988210158401015840 in (22) gives
2119872 = 119872 + 2
2119873 = 119872 +119873
(23)
so that119872 = 119873 = 2 The tanh-coth method admits the use of
119880 (120583120585) = 119878 (119884) =
2
sum
119896=0
119886119896119884119896+
2
sum
119896=1
119887119896119884minus119896
119882 (120583120585) = 119878 (119884) =
2
sum
119896=0
119888119896119884119896+
2
sum
119896=1
119889119896119884119896
(24)
Substituting (24) into (22) collecting the coefficients of 119884and solving the resulting system we find the following setsof solutions
1198860=
1
120573 + 1
1198862=
minus1
2 (120573 + 1)
1198872= 1198862
1198892= plusmn
radic120573 + 2
2 (120573 + 1)
1198882= minus1198892
119881 =
radic3120588120572120573 (120573 + 1)
3120588 (120573 + 1)
120583 =
radic3
119897
1198861= 1198871= 1198880= 1198881= 1198891= 0
1198860=
1
120573 + 1
1198862=
3120573
61205732+ 14120573 + 8
1198872= 1198862
1198892= plusmn
3120573radic120573 + 2
2 (3120573 + 4) (120573 + 1)
1198882= minus1198892
119881 =
radic15120588120572120573 (211989721205832+ 9120573 + 15)
120588 (211989721205832+ 9120573 + 15)
120583 =
radicminus120573 (15120573 + 12)
(5120573 + 4) 119897
1198861= 1198871= 1198880= 1198881= 1198891= 0
1198860=
4 minus 120573
4 (120573 + 1)
1198862=
3120573
4120573 + 1
1198880=
minus1198882
3
1198882= plusmn
9120573
2radic611989721205832+ 36120573 + 18
119881 =
radicminus3120588120572120573 (minus3 + 11989721205832)
120588 (minus3 + 11989721205832)
120583 =
radicminus120573 (3120573 + 6)
(120573 + 2) 119897
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860=
3120573 + 4
4 (120573 + 1)
1198862=
minus3120573
4120573 + 1
1198880= minus1198882
1198882= plusmn
9120573
2radicminus611989721205832+ 36120573 + 18
119881 =
radic3120588120572120573 (3 + 11989721205832)
120588 (3 + 11989721205832)
120583 =
radic3120573 (120573 + 2)
(120573 + 2) 119897
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860=
120573 + 8
8 (120573 + 1)
1198862=
3120573
16 (120573 + 1)
1198872= 1198862 119888
0=
21198892
3
1198882= 1198892 119889
2= plusmn
9120573
8radic2411989721205832+ 36120573 + 18
119881 =
radicminus3120588120572120573 (minus3 + 411989721205832)
120588 (minus3 + 411989721205832)
120583 =
radicminus120573 (3120573 + 6)
2 (120573 + 2) 119897
1198861= 1198871= 1198881= 1198891= 0
1198860=
3 (120573 + 8)
8 (120573 + 1)
1198862=
minus3120573
16 (120573 + 1)
1198872= 1198862 119888
0= minus2119889
2
6 Journal of Applied Mathematics
1198882= 1198892 119889
2= plusmn
9120573
8radicminus2411989721205832+ 36120573 + 18
119881 =
radic3120588120572120573 (3 + 411989721205832)
120588 (3 + 411989721205832)
120583 =
radic120573 (3120573 + 6)
2 (120573 + 2) 119897
1198861= 1198871= 1198881= 1198891= 0
(25)
In accordance with these sets of solutions we have thefollowing solutions
1199061(119909 119905) =
1
120573 + 1
minus
1
2 (120573 + 1)
tanh2radic3
119897
times (119909 minus
radic3120588120572120573 (120573 + 1)
3120588 (120573 + 1)
119905)
minus
1
2 (120573 + 1)
coth2radic3
119897
(119909 minus
radic3120588120572120573 (120573 + 1)
3120588 (120573 + 1)
119905)
1199081(119909 119905) = ∓
radic120573 + 2
2 (120573 + 1)
tanh2radic3
119897
(119909 minus
radic3120588120572120573 (120573 + 1)
3120588 (120573 + 1)
119905)
plusmn
radic120573 + 2
2 (120573 + 1)
coth2radic3
119897
(119909 minus
radic3120588120572120573 (120573 + 1)
3120588 (120573 + 1)
119905)
1199062(119909 119905) =
1
120573 + 1
+
3120573
61205732+ 14120573 + 8
tanh2radicminus120573 (15120573 + 12)
(5120573 + 4) 119897
times (119909 minus
radic15120588120572120573 (211989721205832+ 9120573 + 15)
120588 (211989721205832+ 9120573 + 15)
119905)
+
3120573
61205732+ 14120573 + 8
coth2radicminus120573 (15120573 + 12)
(5120573 + 4) 119897
times (119909 minus
radic15120588120572120573 (211989721205832+ 9120573 + 15)
120588 (211989721205832+ 9120573 + 15)
119905)
1199082(119909 119905) = ∓
3120573radic120573 + 2
2 (3120573 + 4) (120573 + 1)
tanh2radicminus120573 (15120573 + 12)
(5120573 + 4) 119897
times (119909 minus
radic15120588120572120573 (211989721205832+ 9120573 + 15)
120588 (211989721205832+ 9120573 + 15)
119905)
plusmn
3120573radic120573 + 2
2 (3120573 + 4) (120573 + 1)
coth2radicminus120573 (15120573 + 12)
(5120573 + 4) 119897
times (119909 minus
radic15120588120572120573 (211989721205832+ 9120573 + 15)
120588 (211989721205832+ 9120573 + 15)
119905)
1199063(119909 119905) =
4 minus 120573
4 (120573 + 1)
+
3120573
4120573 + 1
tanh2radicminus120573 (3120573 + 6)
(120573 + 2) 119897
times (119909 minus
radicminus3120588120572120573 (minus3 + 11989721205832)
120588 (minus3 + 11989721205832)
119905)
1199083(119909 119905) = ∓
3120573
2radic611989721205832+ 36120573 + 18
plusmn
9120573
2radic611989721205832+ 36120573 + 18
tanh2radicminus120573 (3120573 + 6)
(120573 + 2) 119897
times (119909 minus
radicminus3120588120572120573 (minus3 + 11989721205832)
120588 (minus3 + 11989721205832)
119905)
1199064(119909 119905) =
3120573 + 4
4 (120573 + 1)
minus
3120573
4120573 + 1
tanh2radic3120573 (120573 + 2)
(120573 + 2) 119897
times (119909 minus
radic3120588120572120573 (3 + 11989721205832)
120588 (3 + 11989721205832)
119905)
1199084(119909 119905) = ∓
9120573
2radicminus611989721205832+ 36120573 + 18
plusmn
9120573
2radicminus611989721205832+ 36120573 + 18
tanh2radic3120573 (120573 + 2)
(120573 + 2) 119897
times (119909 minus
radic3120588120572120573 (3 + 11989721205832)
120588 (3 + 11989721205832)
119905)
1199065(119909 119905) =
120573 + 8
8 (120573 + 1)
+
3120573
16 (120573 + 1)
tanh2radicminus120573 (3120573 + 6)
2 (120573 + 2) 119897
Journal of Applied Mathematics 7
times(119909 minus
radicminus3120588120572120573 (minus3 + 411989721205832)
120588 (minus3 + 411989721205832)
119905)
+
3120573
16 (120573 + 1)
coth2radicminus120573 (3120573 + 6)
2 (120573 + 2) 119897
times (119909 minus
radicminus3120588120572120573 (minus3 + 411989721205832)
120588 (minus3 + 411989721205832)
119905)
1199085(119909 119905) = plusmn
3120573
4radic2411989721205832+ 36120573 + 18
plusmn
9120573
8radic2411989721205832+ 36120573 + 18
tanh2radicminus120573 (3120573 + 6)
2 (120573 + 2) 119897
times (119909 minus
radicminus3120588120572120573 (minus3 + 411989721205832)
120588 (minus3 + 411989721205832)
119905)
plusmn
9120573
8radic2411989721205832+ 36120573 + 18
coth2radicminus120573 (3120573 + 6)
2 (120573 + 2) 119897
times (119909 minus
radicminus3120588120572120573 (minus3 + 411989721205832)
120588 (minus3 + 411989721205832)
119905)
1199066(119909 119905) =
3 (120573 + 8)
8 (120573 + 1)
minus
3120573
16 (120573 + 1)
tanh2radic120573 (3120573 + 6)
2 (120573 + 2) 119897
times (119909 minus
radic3120588120572120573 (3 + 411989721205832)
120588 (3 + 411989721205832)
119905)
minus
3120573
16 (120573 + 1)
coth2radic120573 (3120573 + 6)
2 (120573 + 2) 119897
times (119909 minus
radic3120588120572120573 (3 + 411989721205832)
120588 (3 + 411989721205832)
119905)
1199086(119909 119905) = ∓
9120573
4radicminus2411989721205832+ 36120573 + 18
plusmn
9120573
8radicminus2411989721205832+ 36120573 + 18
tanh2radic120573 (3120573 + 6)
2 (120573 + 2) 119897
times (119909 minus
radic3120588120572120573 (3 + 411989721205832)
120588 (3 + 411989721205832)
119905)
plusmn
9120573
8radicminus2411989721205832+ 36120573 + 18
coth2radic120573 (3120573 + 6)
2 (120573 + 2) 119897
times (119909 minus
radic3120588120572120573 (3 + 411989721205832)
120588 (3 + 411989721205832)
119905)
(26)
33 The Third Coupled We now examine the coupled equa-tions
119906119905119905= 120573119906119909119909minus
120573
2
(1199062)119909119909+
1
2
(1199082)119909119909+ 1205722119906119909119909119905119905
119908119905119905= (119906119908)
119909119909+ 1205722119908119909119909119905119905
(27)
Using the wave variable 120585 = 119909 minus119881119905 the system (27) is carriedto a system of ODEs Integrating these equations twice andneglecting constants of integration we find
2 (1198812minus 120573)119880 + 120573119880
2minus 120573119882
2minus 21205722119881211988010158401015840= 0
1198812119882minus119880119882 minus 120572
2119881211988210158401015840= 0
(28)
Balancing 1205731198822 with 21205722119881211988010158401015840 and 119881211988210158401015840 with 119880119882 in (28)gives
2119873 = 119872 + 2
2 + 119873 = 119872 +119873
(29)
so that
119872 = 119873 = 2 (30)
As a result we seek solutions to (27) in the form
119880(120583120585) = 119878 (119884) =
2
sum
119896=0
119886119896119884119896+
2
sum
119896=1
119887119896119884minus119896
119882 (120583120585) = 119878 (119884) =
2
sum
119896=0
119888119896119884119896+
2
sum
119896=1
119889119896119884119896
(31)
Substitution of (31) into (28) leads to an algebraic system inthe unknowns 119886
0 1198861 1198862 1198871 1198872 1198880 1198881 1198882 1198891 1198892 119881 and 120583
8 Journal of Applied Mathematics
Collecting the coefficients of 119884 and solving the resultingsystem we find the following sets of solutions
1198860=
120573 (3120573 + 8)
8 (120573 + 1)
1198862=
minus31205732
16 (120573 + 1)
1198872= 1198862 119888
0= minus2119889
2 119888
2= 1198892
1198892=
plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573
16
119881 =
radic120573 (1612057221205832+ 1)
1612057221205832+ 1
120583 =
radic120573 (120573 + 2)
4120572 (120573 + 2)
1198861= 1198871= 1198881= 1198891= 0
1198860=
120573 (120573 + 8)
8 (120573 + 1)
1198862=
31205732
16 (120573 + 1)
1198872= 1198862
1198880=
21198892
3
1198882= 1198892
1198892=
plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573
16
119881 =
radicminus120573 (1612057221205832minus 1)
1612057221205832minus 1
120583 =
radicminus120573 (120573 + 2)
4120572 (120573 + 2)
1198861= 1198871= 1198881= 1198891= 0
1198860=
minus120573 (120573 minus 4)
4 (120573 + 1)
1198862=
31205732
4 (120573 + 1)
1198880= minus
1198882
3
1198882=
plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573
4
119881 =
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
120583 =
radicminus120573 (120573 + 2)
2120572 (120573 + 2)
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860=
120573 (3120573 + 4)
4 (120573 + 1)
1198862=
minus31205732
4 (120573 + 1)
1198880= minus2
1198882=
plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573
4
119881 =
radic120573 (412057221205832+ 1)
412057221205832+ 1
120583 =
radic120573 (120573 + 2)
2120572 (120573 + 2)
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860=
120573
120573 + 1
1198862=
31205732
2 (31205732+ 7120573 + 4)
1198872= 1198862 119888
2= minus1198892
1198892=
plusmn3radicminus421198814120573 minus 58119881
4minus 21198812120573 minus 120119881
2+ 120120573
8
119881 =
radic5120573 (812057221205832+ 3120573 + 5)
812057221205832+ 3120573 + 5
120583 =
radicminus120573 (5120573 + 4)
2120572 (5120573 + 4)
1198861= 1198871= 1198880= 1198881= 1198891= 0
1198860=
120573
120573 + 1
1198862=
minus120573
2 (120573 + 1)
1198872= 1198862
1198882= minus1198892 119889
2=
plusmnradic91198812+ 6
2
119881 =
radic3120573 (120573 + 1)
3 (120573 + 1)
120583 =
minus1
2120572
1198861= 1198871= 1198880= 1198881= 1198891= 0
(32)
Consequently we obtain the following solutions
1199061(119909 119905) =
120573 (3120573 + 8)
8 (120573 + 1)
minus
31205732
16 (120573 + 1)
tanh2radic120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radic120573 (1612057221205832+ 1)
1612057221205832+ 1
119905)
minus
31205732
16 (120573 + 1)
coth2radic120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radic120573 (1612057221205832+ 1)
1612057221205832+ 1
119905)
1199081(119909 119905) =
∓3radicminus41198814minus 41198812+ 21205732+ 4120573
8
plusmn
3radicminus41198814minus 41198812+ 21205732+ 4120573
16
tanh2radic120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radic120573 (1612057221205832+ 1)
1612057221205832+ 1
119905)
plusmn
3radicminus41198814minus 41198812+ 21205732+ 4120573
16
coth2radic120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radic120573 (1612057221205832+ 1)
1612057221205832+ 1
119905)
Journal of Applied Mathematics 9
1199062(119909 119905) =
120573 (120573 + 8)
8 (120573 + 1)
+
31205732
16 (120573 + 1)
tanh2radicminus120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radicminus120573 (1612057221205832minus 1)
1612057221205832minus 1
119905)
+
31205732
16 (120573 + 1)
coth2radicminus120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radicminus120573 (1612057221205832minus 1)
1612057221205832minus 1
119905)
1199082(119909 119905) = plusmn
radicminus41198814minus 41198812+ 21205732+ 4120573
8
plusmn
3radicminus41198814minus 41198812+ 21205732+ 4120573
16
tanh2radicminus120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radicminus120573 (1612057221205832minus 1)
1612057221205832minus 1
119905)
plusmn
3radicminus41198814minus 41198812+ 21205732+ 4120573
16
coth2radicminus120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radicminus120573 (1612057221205832minus 1)
1612057221205832minus 1
119905)
1199063(119909 119905) =
minus120573 (120573 minus 4)
4 (120573 + 1)
+
31205732
4 (120573 + 1)
tanh2radicminus120573 (120573 + 2)
2120572 (120573 + 2)
times (119909 minus
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
119905)
1199083(119909 119905) = ∓
radicminus41198814minus 41198812+ 21205732+ 4120573
4
plusmn
3radicminus41198814minus 41198812+ 21205732+ 4120573
4
tanhradicminus120573 (120573 + 2)
2120572 (120573 + 2)
times (119909 minus
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
119905)
1199064(119909 119905) =
120573 (3120573 + 4)
4 (120573 + 1)
minus
31205732
4 (120573 + 1)
tanh2radic120573 (120573 + 2)
2120572 (120573 + 2)
times (119909 minus
radic120573 (412057221205832+ 1)
412057221205832+ 1
119905)
1199084(119909 119905) = ∓
3radicminus41198814minus 41198812+ 21205732+ 4120573
4
plusmn
3radicminus41198814minus 41198812+ 21205732+ 4120573
4
tanh2radic120573 (120573 + 2)
2120572 (120573 + 2)
times (119909 minus
radic120573 (412057221205832+ 1)
412057221205832+ 1
119905)
1199065(119909 119905) =
120573
120573 + 1
+
31205732
2 (31205732+ 7120573 + 4)
tanh2radicminus120573 (5120573 + 4)
2120572 (5120573 + 4)
times (119909 minus
radic5120573 (812057221205832+ 3120573 + 5)
812057221205832+ 3120573 + 5
119905)
+
31205732
2 (31205732+ 7120573 + 4)
coth2radicminus120573 (5120573 + 4)
2120572 (5120573 + 4)
times (119909 minus
radic5120573 (812057221205832+ 3120573 + 5)
812057221205832+ 3120573 + 5
119905)
1199085(119909 119905)
= ∓
3radicminus421198814120573 minus 58119881
4minus 21198812120573 minus 120119881
2+ 120120573
8
times tanh2radicminus120573 (5120573 + 4)
2120572 (5120573 + 4)
times (119909 minus
radic5120573 (812057221205832+ 3120573 + 5)
812057221205832+ 3120573 + 5
119905)
plusmn
3radicminus421198814120573 minus 58119881
4minus 21198812120573 minus 120119881
2+ 120120573
8
10 Journal of Applied Mathematics
times coth2radicminus120573 (5120573 + 4)
2120572 (5120573 + 4)
times (119909 minus
radic5120573 (812057221205832+ 3120573 + 5)
812057221205832+ 3120573 + 5
119905)
1199066(119909 119905) =
120573
120573 + 1
minus
120573
2 (120573 + 1)
tanh2minus12120572
(119909 minus
radic3120573 (120573 + 1)
3 (120573 + 1)
119905)
minus
120573
2 (120573 + 1)
coth2minus12120572
(119909 minus
radic3120573 (120573 + 1)
3 (120573 + 1)
119905)
1199086(119909 119905)
= ∓
radic91198812+ 6
2
tanh2minus12120572
(119909 minus
radic3120573 (120573 + 1)
3 (120573 + 1)
119905)
plusmn
radic91198812+ 6
2
coth2minus12120572
(119909 minus
radic3120573 (120573 + 1)
3 (120573 + 1)
119905)
(33)
34 The Fourth Coupled We now consider the coupled IBqequations
119906119905119905= 119906119909119909minus 120573(119906
2)119909119909+ (1199082)119909119909+ 119906119909119909119905119905
119908119905119905= 119908119909119909+ 2(119906119908)
119909119909+ 119908119909119909119905119905
(34)
Using the wave variable 120585 = 119909 minus 119881119905 (34) can be converted toa system of ODEs
(1198812minus 1)119880 + 120573119880
2minus1198822minus 119881211988010158401015840= 0
(1198812minus 1)119882 minus 2119880119882 minus 119881
211988210158401015840= 0
(35)
Balancing1198822 with 11988010158401015840 and 119880119882 with11988210158401015840 in (35) gives
2119873 = 119872 + 2
119872 + 119873 = 119873 + 2
(36)
so that
119872 = 119873 = 2 (37)
The tanh-coth method substitutes the finite expansions
119880(120583120585) = 119878 (119884) =
2
sum
119896=0
119886119896119884119896+
2
sum
119896=1
119887119896119884minus119896
119882 (120583120585) = 119878 (119884) =
2
sum
119896=0
119888119896119884119896+
2
sum
119896=1
119889119896119884119896
(38)
In (35) collecting the coefficients of each power of 119884 andsolving the resulting system we obtain
1198860=
minus21205832
161205832+ 1
1198862=
minus31205832
161205832+ 1
1198862= 1198872
1198880=
21198892
3
1198882= 1198892
1198892=
plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
119881 =
1
radic161205832+ 1
1198861= 1198871= 1198881= 1198891= 0
1198860=
minus61205832
161205832minus 1
1198862=
31205832
161205832minus 1
1198862= 1198872
1198880= minus2119889
2 119888
2= 1198892
1198892=
plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
119881 =
1
radicminus161205832+ 1
1198861= 1198871= 1198881= 1198891= 0
1198860=
1205832
41205832+ 1
1198862=
minus31205832
41205832+ 1
1198880=
minus1198882
3
1198882=
plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
119881 =
1
radic41205832+ 1
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860=
minus31205832
41205832minus 1
1198862=
31205832
41205832minus 1
1198880= minus1198882
1198882=
plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
119881 =
1
radicminus41205832+ 1
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860=
minus31205832
41205832minus 1
1198872=
31205832
41205832minus 1
1198880= minus1198892
1198892=
plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
119881 =
1
radicminus41205832+ 1
1198861= 1198862= 1198871= 1198881= 1198882= 1198891= 0
(39)
Journal of Applied Mathematics 11
where 120583 is left as a free parameter Consequently we obtainthe following travelling wave solutions
1199061(119909 119905) = minus
21205832
161205832+ 1
minus
31205832
161205832+ 1
tanh2120583(119909 minus 1
radic161205832+ 1
119905)
minus
31205832
161205832+ 1
coth2120583(119909 minus 1
radic161205832+ 1
119905)
1199081(119909 119905) = plusmn
(119881 + 1) (119881 minus 1)radic120573 + 2
8
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
tanh2120583
times(119909 minus
1
radic161205832+ 1
119905)
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
coth2120583
times(119909 minus
1
radic161205832+ 1
119905)
1199062(119909 119905) = minus
61205832
161205832minus 1
+
31205832
161205832minus 1
tanh2120583(119909 minus 1
radicminus161205832+ 1
119905)
+
31205832
161205832minus 1
coth2120583(119909 minus 1
radicminus161205832+ 1
119905)
1199082(119909 119905) = ∓
3 (119881 + 1) (119881 minus 1)radic120573 + 2
8
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
tanh2120583
times(119909 minus
1
radicminus161205832+ 1
119905)
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
coth2120583
times(119909 minus
1
radicminus161205832+ 1
119905)
1199063(119909 119905) =
1205832
41205832+ 1
minus
31205832
41205832+ 1
tanh2120583(119909 minus 1
radic41205832+ 1
119905)
1199083(119909 119905) = ∓
(119881 + 1) (119881 minus 1)radic120573 + 2
4
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
times tanh2120583(119909 minus 1
radic41205832+ 1
119905)
1199064(119909 119905) = minus
31205832
41205832minus 1
+
31205832
41205832minus 1
tanh2120583(119909 minus 1
radicminus41205832+ 1
119905)
1199084(119909 119905) = ∓
3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
times tanh2120583(119909 minus 1
radicminus41205832+ 1
119905)
1199065(119909 119905) = minus
31205832
41205832minus 1
+
31205832
41205832minus 1
coth2120583(119909 minus 1
radicminus41205832+ 1
119905)
1199085(119909 119905) = ∓
3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
plusmn 3
(119881 + 1) (119881 minus 1)radic120573 + 2
4
times coth2120583(119909 minus 1
radicminus41205832+ 1
119905)
(40)
35 The Fifth Coupled We finally consider the coupled IBqequations
119906119905119905= 120573119906119909119909minus
120573
2
(1199062)119909119909+
120573
2
(1199082)119909119909+ 1205722119906119909119909119905119905
119908119905119905= 120574(119906119908)
119909119909+ 120588(119908
2)119909119909+ 1205722119908119909119909119905119905
(41)
12 Journal of Applied Mathematics
This system of equations (41) can be converted to the follow-ing system of ODEs by using the wave variable 120585 = 119909 minus 119881119905
(21198812minus 2120573)119880 + 120573119880
2minus 120573119882
2minus 21205722119881211988010158401015840= 0
1198812119882minus 120574119880119882 minus 120588119882
2minus 1205722119881211988210158401015840= 0
(42)
Using the balancing procedure in (42) gives
2119872 = 119872 +119873
2119873 = 119872 +119873
(43)
so that
119872 = 119873 = 2 (44)
We consider solutions in the following form
119880 (120583120585) = 119878 (119884) =
2
sum
119896=0
119886119896119884119896+
2
sum
119896=1
119887119896119884minus119896
119882 (120583120585) = 119878 (119884) =
2
sum
119896=0
119888119896119884119896+
2
sum
119896=1
119889119896119884119896
(45)
Substituting (45) into (42) collecting the coefficients of 119884and solving the resulting system we find the following setsof solutions
1198860= (minus120573 (8120588119888
21205732120574 minus 8120573120574
21205881198882minus 18120588
21205732+ 24120573120588
31198882
+1212057321205742minus 16120574
31205881198882+ 24120588
31198882120574 minus 3120574120573
3))
times (4 (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882+ 312058821205733
+ 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882minus 312057331205742minus 312057431205732))
minus1
1198862=
minus (41205881198882120573 + 4119888
2120588120574 minus 3120573
2)
4120574 (120583 + 120573)
1198880= (120573(120588
212057321198882+ 120574211988821205732+ 12057311988821205743+ 2120574212058821198882
+312057412058821205731198882minus
3
2
1205881205732120574 minus
3
4
1205881205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882+ 312058821205733
+ 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882minus 312057331205742minus 312057431205732)
minus1
1198882= (plusmn3 (minus 120573120588 + (minus120588
21205732+ 41205743120573 minus 4120574
21198814
minus 412057431198812+ 212057321205742
minus41205882120574120573 + 4120588
21198814+ 412058821198812120574)
12
))
times (4 (1205742minus 1205882))
minus1
119881 =
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
120583 =
radicminus120573 (2120574 + 120573)
2120572 (2120574 + 120573)
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860= (120573(6120588119888
21205732120574 + 10120573120574
21205881198882+
3
2
12058821205732+ 2120573120588
31198882
+312057321205742+ 412057431205881198882+ 212058831198882120574 +
9
4
1205741205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882minus 312058821205733
minus 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882+ 312057331205742+ 312057431205732)
minus1
1198862=
minus (41205881198882120573 + 4119888
2120588120574 + 3120573
2)
4120574 (120583 + 120573)
1198880= (120573(minus 3120588
212057321198882minus 3120574211988821205732minus 312057311988821205743
minus6120574212058821198882minus 9120574120588
21205731198882minus
9
2
1205881205732120574 minus
9
4
1205881205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882
minus 312058821205733minus 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882+ 312057331205742+ 312057431205732)
minus1
1198882= (plusmn3 (120573120588 + (minus120588
21205732+ 41205743120573 minus 4120574
21198814
minus 412057431198812+ 212057321205742minus 41205882120574120573
+412058821198814+ 412058821198812120574)
12
))
times (4 (1205742minus 1205882))
minus1
119881 =
radic120573 (412057221205832+ 1)
412057221205832+ 1
120583 =
radic120573 (2120574 + 120573)
2120572 (2120574 + 120573)
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
(46)
Journal of Applied Mathematics 13
In view of this we obtain the following travelling wavesolutions
1199061(119909 119905) = (minus120573 (8120588119888
21205732120574 minus 8120573120574
21205881198882minus 18120588
21205732+ 24120573120588
31198882
+1212057321205742minus 16120574
31205881198882+ 24120588
31198882120574 minus 3120574120573
3))
times (4 (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882+ 312058821205733
+ 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882minus 312057331205742minus 312057431205732))
minus1
minus
(41205881198882120573 + 4119888
2120588120574 minus 3120573
2)
4120574 (120583 + 120573)
tanh2radicminus120573 (2120574 + 120573)
2120572 (2120574 + 120573)
times(119909 minus
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
119905)
1199081(119909 119905) = (120573 (120588
212057321198882+ 120574211988821205732+ 12057311988821205743+ 2120574212058821198882
+312057412058821205731198882minus
3
2
1205881205732120574 minus
3
4
1205881205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882+ 312058821205733
+ 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882minus 312057331205742minus 312057431205732)
minus1
plusmn (3 (minus 120573120588 + (minus12058821205732+ 41205743120573 minus 4120574
21198814
minus 412057431198812+ 212057321205742minus 41205882120574120573
+412058821198814+ 412058821198812120574)
12
))
times (4 (1205742minus 1205882))
minus1
times tanh2radicminus120573 (2120574 + 120573)
2120572 (2120574 + 120573)
times(119909 minus
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
119905)
1199062(119909 119905) = (120573 (6120588119888
21205732120574 + 10120573120574
21205881198882+
3
2
12058821205732+ 2120573120588
31198882
+312057321205742+ 412057431205881198882+ 212058831198882120574 +
9
4
1205741205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882minus 312058821205733
minus 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882+ 312057331205742+ 312057431205732)
minus1
minus
(41205881198882120573 + 4119888
2120588120574 + 3120573
2)
4120574 (120583 + 120573)
tanh2radic120573 (2120574 + 120573)
2120572 (2120574 + 120573)
times(119909 minus
radic120573 (412057221205832+ 1)
412057221205832+ 1
119905)
1199082(119909 119905) = (120573 (minus 3120588
212057321198882minus 3120574211988821205732minus 312057311988821205743
minus6120574212058821198882minus 9120574120588
21205731198882minus
9
2
1205881205732120574 minus
9
4
1205881205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882minus 312058821205733
minus 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882+ 312057331205742+ 312057431205732)
minus1
plusmn (3 (120573120588 + (minus12058821205732+ 41205743120573 minus 4120574
21198814
minus 412057431198812+ 212057321205742minus 41205882120574120573
+412058821198814+ 412058821198812120574)
12
))
times (4 (1205742minus 1205882))
minus1
times tanh2radic120573 (2120574 + 120573)
2120572 (2120574 + 120573)
(119909 minus
radic120573 (412057221205832+ 1)
412057221205832+ 1
119905)
(47)
4 Conclusion
In summary the tanh-coth method has been successfullyimplemented to find new travelling wave solutions for severalkinds of two coupled improved Boussinesq equations Thesolutions that have been found can be seen as an expansionand verification of the previously known data However thisstudy confirmed again the belief that the tanh-coth methodis a powerful and reliable technique to handle nonlineardispersive-dissipative equations and system of equationsThroughout the work Maple has been used to overcome thetedious algebraic calculations
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] A De Godefroy ldquoBlow up of solutions of a generalized Boussi-nesq equationrdquo IMA Journal of AppliedMathematics vol 60 no2 pp 123ndash138 1998
14 Journal of Applied Mathematics
[2] S Wang and M Li ldquoThe Cauchy problem for coupled IMBqequationsrdquo IMA Journal of Applied Mathematics vol 74 no 5pp 726ndash740 2009
[3] N Duruk H A Erbay and A Erkip ldquoBlow-up and globalexistence for a general class of nonlocal nonlinear coupled waveequationsrdquo Journal of Differential Equations vol 250 no 3 pp1448ndash1459 2011
[4] G Chen and H Zhang ldquoInitial boundary value problem for asystem of generalized IMBq equationsrdquoMathematical Methodsin the Applied Sciences vol 27 no 5 pp 497ndash518 2004
[5] P Rosenau ldquoDynamics of dense latticesrdquo Physical Review B vol36 no 11 pp 5868ndash5876 1987
[6] P L Christiansen P S Lomdahl andVMuto ldquoOn aToda latticemodel with a transversal degree of freedomrdquo Nonlinearity vol4 no 2 pp 477ndash501 1991
[7] S K Turitsyn ldquoOn a Toda lattice model with a transversaldegree of freedom Sufficient criterion of blow-up in thecontinuum limitrdquo Physics Letters A vol 173 no 3 pp 267ndash2691993
[8] R L Pego P Smereka and M I Weinstein ldquoOscillatoryinstability of solitary waves in a continuum model of latticevibrationsrdquo Nonlinearity vol 8 no 6 pp 921ndash941 1995
[9] X Fan and L Tian ldquoCompacton solutions and multiple com-pacton solutions for a continuum Toda lattice modelrdquo ChaosSolitons and Fractals vol 29 no 4 pp 882ndash894 2006
[10] A M Wazwaz ldquoThe Hirotarsquos bilinear method and the tanh-coth method for multiple-soliton solutions of the Sawada-Kotera-Kadomtsev-Petviashvili equationrdquoAppliedMathematicsand Computation vol 200 no 1 pp 160ndash166 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Journal of Applied Mathematics
Consequently we obtain the following travelling wave solu-tions
1199061(119909 119905) = minus
radic120572120573 (1198814minus 31198812+ 2)
41205722
+
3radic120572120573 (1198814minus 31198812+ 2)
41205722
tanh2radic3120573 (minus3120573 + 4120572)
minus3120573 + 4120572
times(119909 minus
6
radicminus61205832+ 18
119905)
1199081(119909 119905) =
120573 + 4120572
8120572 (120572 minus 120573)
minus
3120573
8120572 (120572 minus 120573)
tanh2radic3120573 (minus3120573 + 4120572)
minus3120573 + 4120572
times(119909 minus
6
radicminus61205832+ 18
119905)
1199062(119909 119905) = minus
3radic120572120573 (1198814minus 31198812+ 2)
81205722
+
91205732(2120572 minus 120573)
102412057231198872(120572 minus 120573)
2tanh2
radic120573 (9120573 minus 12120572)
minus6120573 + 8120572
times(119909 minus
6
radic241205832+ 18
119905)
+
3radic120572120573 (1198814minus 31198812+ 2)
161205722
coth2radic120573 (9120573 minus 12120572)
minus6120573 + 8120572
times(119909 minus
6
radic241205832+ 18
119905)
1199082(119909 119905) =
minus3120573 + 8120572
16120572 (120572 minus 120573)
+
3120573
32120572 (120572 minus 120573)
tanh2radic120573 (9120573 minus 12120572)
minus6120573 + 8120572
times(119909 minus
6
radic241205832+ 18
119905)
+
3120573
32120572 (120572 minus 120573)
coth2radic120573 (9120573 minus 12120572)
minus6120573 + 8120572
times(119909 minus
6
radic241205832+ 18
119905)
1199063(119909 119905) =
radic120572120573 (1198814minus 31198812+ 2)
81205722
+
91205732(2120572 minus 120573)
102412057231198872(120572 minus 120573)
2tanh2
radic3120573 (minus3120573 + 4120572)
minus6120573 + 8120572
times(119909 minus
6
radicminus241205832+ 18
119905)
+
3radic120572120573 (1198814minus 31198812+ 2)
161205722
coth2radic3120573 (minus3120573 + 4120572)
minus6120573 + 8120572
times(119909 minus
6
radicminus241205832+ 18
119905)
1199083(119909 119905) =
minus120573 + 8120572
16120572 (120572 minus 120573)
minus
3120573
32120572 (120572 minus 120573)
tanh2radic3120573 (minus3120573 + 4120572)
minus6120573 + 8120572
times(119909 minus
6
radicminus241205832+ 18
119905)
minus
3120573
32120572 (120572 minus 120573)
coth2radic3120573 (minus3120573 + 4120572)
minus6120573 + 8120572
times(119909 minus
6
radicminus241205832+ 18
119905)
1199064(119909 119905) =
minus91205732(2120572 minus 120573)
6412057231198862(120572 minus 120573)
2
+
3radic120572120573 (1198814minus 31198812+ 2)
41205722
tanh2radic120573 (9120573 minus 12120572)
minus3120573 + 4120572
times(119909 minus
6
radic61205832+ 18
119905)
Journal of Applied Mathematics 5
1199084(119909 119905) =
minus3120573 + 4120572
8120572 (120572 minus 120573)
+
3120573
8120572 (120572 minus 120573)
tanh2radic120573 (9120573 minus 12120572)
minus3120573 + 4120572
times(119909 minus
6
radic61205832+ 18
119905)
(20)
32 The Second Coupled We next consider the system ofequations
120588
120572
119906119905119905= 120573119906119909119909minus
1205732
2
(1199062)119909119909+
120573
2
(1199082)119909119909+
120588
120572
1198972
12
119906119909119909119905119905
120588
120572
119908119905119905= 120573(119906119908)
119909119909+
120588
120572
1198972
12
119908119909119909119905119905
(21)
Using thewave variable 120585 = 119909minus119881119905 and then by integrating thisequation twice and considering the constants of integration tobe zero the system (21) is carried to a system of ODEs
12 (120572120573 minus 1205881198812)119880 minus 6120572120573
21198802+ 6120572120573119882
2+ 1205881198972119881211988010158401015840= 0
12120572120573119880119882 + 1205881198972119881211988210158401015840minus 12120588119881
2119882 = 0
(22)
Balancing minus612057212057321198802 with 120588119897
2119881211988010158401015840 and 12120572120573119880119882 with
1205881198972119881211988210158401015840 in (22) gives
2119872 = 119872 + 2
2119873 = 119872 +119873
(23)
so that119872 = 119873 = 2 The tanh-coth method admits the use of
119880 (120583120585) = 119878 (119884) =
2
sum
119896=0
119886119896119884119896+
2
sum
119896=1
119887119896119884minus119896
119882 (120583120585) = 119878 (119884) =
2
sum
119896=0
119888119896119884119896+
2
sum
119896=1
119889119896119884119896
(24)
Substituting (24) into (22) collecting the coefficients of 119884and solving the resulting system we find the following setsof solutions
1198860=
1
120573 + 1
1198862=
minus1
2 (120573 + 1)
1198872= 1198862
1198892= plusmn
radic120573 + 2
2 (120573 + 1)
1198882= minus1198892
119881 =
radic3120588120572120573 (120573 + 1)
3120588 (120573 + 1)
120583 =
radic3
119897
1198861= 1198871= 1198880= 1198881= 1198891= 0
1198860=
1
120573 + 1
1198862=
3120573
61205732+ 14120573 + 8
1198872= 1198862
1198892= plusmn
3120573radic120573 + 2
2 (3120573 + 4) (120573 + 1)
1198882= minus1198892
119881 =
radic15120588120572120573 (211989721205832+ 9120573 + 15)
120588 (211989721205832+ 9120573 + 15)
120583 =
radicminus120573 (15120573 + 12)
(5120573 + 4) 119897
1198861= 1198871= 1198880= 1198881= 1198891= 0
1198860=
4 minus 120573
4 (120573 + 1)
1198862=
3120573
4120573 + 1
1198880=
minus1198882
3
1198882= plusmn
9120573
2radic611989721205832+ 36120573 + 18
119881 =
radicminus3120588120572120573 (minus3 + 11989721205832)
120588 (minus3 + 11989721205832)
120583 =
radicminus120573 (3120573 + 6)
(120573 + 2) 119897
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860=
3120573 + 4
4 (120573 + 1)
1198862=
minus3120573
4120573 + 1
1198880= minus1198882
1198882= plusmn
9120573
2radicminus611989721205832+ 36120573 + 18
119881 =
radic3120588120572120573 (3 + 11989721205832)
120588 (3 + 11989721205832)
120583 =
radic3120573 (120573 + 2)
(120573 + 2) 119897
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860=
120573 + 8
8 (120573 + 1)
1198862=
3120573
16 (120573 + 1)
1198872= 1198862 119888
0=
21198892
3
1198882= 1198892 119889
2= plusmn
9120573
8radic2411989721205832+ 36120573 + 18
119881 =
radicminus3120588120572120573 (minus3 + 411989721205832)
120588 (minus3 + 411989721205832)
120583 =
radicminus120573 (3120573 + 6)
2 (120573 + 2) 119897
1198861= 1198871= 1198881= 1198891= 0
1198860=
3 (120573 + 8)
8 (120573 + 1)
1198862=
minus3120573
16 (120573 + 1)
1198872= 1198862 119888
0= minus2119889
2
6 Journal of Applied Mathematics
1198882= 1198892 119889
2= plusmn
9120573
8radicminus2411989721205832+ 36120573 + 18
119881 =
radic3120588120572120573 (3 + 411989721205832)
120588 (3 + 411989721205832)
120583 =
radic120573 (3120573 + 6)
2 (120573 + 2) 119897
1198861= 1198871= 1198881= 1198891= 0
(25)
In accordance with these sets of solutions we have thefollowing solutions
1199061(119909 119905) =
1
120573 + 1
minus
1
2 (120573 + 1)
tanh2radic3
119897
times (119909 minus
radic3120588120572120573 (120573 + 1)
3120588 (120573 + 1)
119905)
minus
1
2 (120573 + 1)
coth2radic3
119897
(119909 minus
radic3120588120572120573 (120573 + 1)
3120588 (120573 + 1)
119905)
1199081(119909 119905) = ∓
radic120573 + 2
2 (120573 + 1)
tanh2radic3
119897
(119909 minus
radic3120588120572120573 (120573 + 1)
3120588 (120573 + 1)
119905)
plusmn
radic120573 + 2
2 (120573 + 1)
coth2radic3
119897
(119909 minus
radic3120588120572120573 (120573 + 1)
3120588 (120573 + 1)
119905)
1199062(119909 119905) =
1
120573 + 1
+
3120573
61205732+ 14120573 + 8
tanh2radicminus120573 (15120573 + 12)
(5120573 + 4) 119897
times (119909 minus
radic15120588120572120573 (211989721205832+ 9120573 + 15)
120588 (211989721205832+ 9120573 + 15)
119905)
+
3120573
61205732+ 14120573 + 8
coth2radicminus120573 (15120573 + 12)
(5120573 + 4) 119897
times (119909 minus
radic15120588120572120573 (211989721205832+ 9120573 + 15)
120588 (211989721205832+ 9120573 + 15)
119905)
1199082(119909 119905) = ∓
3120573radic120573 + 2
2 (3120573 + 4) (120573 + 1)
tanh2radicminus120573 (15120573 + 12)
(5120573 + 4) 119897
times (119909 minus
radic15120588120572120573 (211989721205832+ 9120573 + 15)
120588 (211989721205832+ 9120573 + 15)
119905)
plusmn
3120573radic120573 + 2
2 (3120573 + 4) (120573 + 1)
coth2radicminus120573 (15120573 + 12)
(5120573 + 4) 119897
times (119909 minus
radic15120588120572120573 (211989721205832+ 9120573 + 15)
120588 (211989721205832+ 9120573 + 15)
119905)
1199063(119909 119905) =
4 minus 120573
4 (120573 + 1)
+
3120573
4120573 + 1
tanh2radicminus120573 (3120573 + 6)
(120573 + 2) 119897
times (119909 minus
radicminus3120588120572120573 (minus3 + 11989721205832)
120588 (minus3 + 11989721205832)
119905)
1199083(119909 119905) = ∓
3120573
2radic611989721205832+ 36120573 + 18
plusmn
9120573
2radic611989721205832+ 36120573 + 18
tanh2radicminus120573 (3120573 + 6)
(120573 + 2) 119897
times (119909 minus
radicminus3120588120572120573 (minus3 + 11989721205832)
120588 (minus3 + 11989721205832)
119905)
1199064(119909 119905) =
3120573 + 4
4 (120573 + 1)
minus
3120573
4120573 + 1
tanh2radic3120573 (120573 + 2)
(120573 + 2) 119897
times (119909 minus
radic3120588120572120573 (3 + 11989721205832)
120588 (3 + 11989721205832)
119905)
1199084(119909 119905) = ∓
9120573
2radicminus611989721205832+ 36120573 + 18
plusmn
9120573
2radicminus611989721205832+ 36120573 + 18
tanh2radic3120573 (120573 + 2)
(120573 + 2) 119897
times (119909 minus
radic3120588120572120573 (3 + 11989721205832)
120588 (3 + 11989721205832)
119905)
1199065(119909 119905) =
120573 + 8
8 (120573 + 1)
+
3120573
16 (120573 + 1)
tanh2radicminus120573 (3120573 + 6)
2 (120573 + 2) 119897
Journal of Applied Mathematics 7
times(119909 minus
radicminus3120588120572120573 (minus3 + 411989721205832)
120588 (minus3 + 411989721205832)
119905)
+
3120573
16 (120573 + 1)
coth2radicminus120573 (3120573 + 6)
2 (120573 + 2) 119897
times (119909 minus
radicminus3120588120572120573 (minus3 + 411989721205832)
120588 (minus3 + 411989721205832)
119905)
1199085(119909 119905) = plusmn
3120573
4radic2411989721205832+ 36120573 + 18
plusmn
9120573
8radic2411989721205832+ 36120573 + 18
tanh2radicminus120573 (3120573 + 6)
2 (120573 + 2) 119897
times (119909 minus
radicminus3120588120572120573 (minus3 + 411989721205832)
120588 (minus3 + 411989721205832)
119905)
plusmn
9120573
8radic2411989721205832+ 36120573 + 18
coth2radicminus120573 (3120573 + 6)
2 (120573 + 2) 119897
times (119909 minus
radicminus3120588120572120573 (minus3 + 411989721205832)
120588 (minus3 + 411989721205832)
119905)
1199066(119909 119905) =
3 (120573 + 8)
8 (120573 + 1)
minus
3120573
16 (120573 + 1)
tanh2radic120573 (3120573 + 6)
2 (120573 + 2) 119897
times (119909 minus
radic3120588120572120573 (3 + 411989721205832)
120588 (3 + 411989721205832)
119905)
minus
3120573
16 (120573 + 1)
coth2radic120573 (3120573 + 6)
2 (120573 + 2) 119897
times (119909 minus
radic3120588120572120573 (3 + 411989721205832)
120588 (3 + 411989721205832)
119905)
1199086(119909 119905) = ∓
9120573
4radicminus2411989721205832+ 36120573 + 18
plusmn
9120573
8radicminus2411989721205832+ 36120573 + 18
tanh2radic120573 (3120573 + 6)
2 (120573 + 2) 119897
times (119909 minus
radic3120588120572120573 (3 + 411989721205832)
120588 (3 + 411989721205832)
119905)
plusmn
9120573
8radicminus2411989721205832+ 36120573 + 18
coth2radic120573 (3120573 + 6)
2 (120573 + 2) 119897
times (119909 minus
radic3120588120572120573 (3 + 411989721205832)
120588 (3 + 411989721205832)
119905)
(26)
33 The Third Coupled We now examine the coupled equa-tions
119906119905119905= 120573119906119909119909minus
120573
2
(1199062)119909119909+
1
2
(1199082)119909119909+ 1205722119906119909119909119905119905
119908119905119905= (119906119908)
119909119909+ 1205722119908119909119909119905119905
(27)
Using the wave variable 120585 = 119909 minus119881119905 the system (27) is carriedto a system of ODEs Integrating these equations twice andneglecting constants of integration we find
2 (1198812minus 120573)119880 + 120573119880
2minus 120573119882
2minus 21205722119881211988010158401015840= 0
1198812119882minus119880119882 minus 120572
2119881211988210158401015840= 0
(28)
Balancing 1205731198822 with 21205722119881211988010158401015840 and 119881211988210158401015840 with 119880119882 in (28)gives
2119873 = 119872 + 2
2 + 119873 = 119872 +119873
(29)
so that
119872 = 119873 = 2 (30)
As a result we seek solutions to (27) in the form
119880(120583120585) = 119878 (119884) =
2
sum
119896=0
119886119896119884119896+
2
sum
119896=1
119887119896119884minus119896
119882 (120583120585) = 119878 (119884) =
2
sum
119896=0
119888119896119884119896+
2
sum
119896=1
119889119896119884119896
(31)
Substitution of (31) into (28) leads to an algebraic system inthe unknowns 119886
0 1198861 1198862 1198871 1198872 1198880 1198881 1198882 1198891 1198892 119881 and 120583
8 Journal of Applied Mathematics
Collecting the coefficients of 119884 and solving the resultingsystem we find the following sets of solutions
1198860=
120573 (3120573 + 8)
8 (120573 + 1)
1198862=
minus31205732
16 (120573 + 1)
1198872= 1198862 119888
0= minus2119889
2 119888
2= 1198892
1198892=
plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573
16
119881 =
radic120573 (1612057221205832+ 1)
1612057221205832+ 1
120583 =
radic120573 (120573 + 2)
4120572 (120573 + 2)
1198861= 1198871= 1198881= 1198891= 0
1198860=
120573 (120573 + 8)
8 (120573 + 1)
1198862=
31205732
16 (120573 + 1)
1198872= 1198862
1198880=
21198892
3
1198882= 1198892
1198892=
plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573
16
119881 =
radicminus120573 (1612057221205832minus 1)
1612057221205832minus 1
120583 =
radicminus120573 (120573 + 2)
4120572 (120573 + 2)
1198861= 1198871= 1198881= 1198891= 0
1198860=
minus120573 (120573 minus 4)
4 (120573 + 1)
1198862=
31205732
4 (120573 + 1)
1198880= minus
1198882
3
1198882=
plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573
4
119881 =
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
120583 =
radicminus120573 (120573 + 2)
2120572 (120573 + 2)
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860=
120573 (3120573 + 4)
4 (120573 + 1)
1198862=
minus31205732
4 (120573 + 1)
1198880= minus2
1198882=
plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573
4
119881 =
radic120573 (412057221205832+ 1)
412057221205832+ 1
120583 =
radic120573 (120573 + 2)
2120572 (120573 + 2)
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860=
120573
120573 + 1
1198862=
31205732
2 (31205732+ 7120573 + 4)
1198872= 1198862 119888
2= minus1198892
1198892=
plusmn3radicminus421198814120573 minus 58119881
4minus 21198812120573 minus 120119881
2+ 120120573
8
119881 =
radic5120573 (812057221205832+ 3120573 + 5)
812057221205832+ 3120573 + 5
120583 =
radicminus120573 (5120573 + 4)
2120572 (5120573 + 4)
1198861= 1198871= 1198880= 1198881= 1198891= 0
1198860=
120573
120573 + 1
1198862=
minus120573
2 (120573 + 1)
1198872= 1198862
1198882= minus1198892 119889
2=
plusmnradic91198812+ 6
2
119881 =
radic3120573 (120573 + 1)
3 (120573 + 1)
120583 =
minus1
2120572
1198861= 1198871= 1198880= 1198881= 1198891= 0
(32)
Consequently we obtain the following solutions
1199061(119909 119905) =
120573 (3120573 + 8)
8 (120573 + 1)
minus
31205732
16 (120573 + 1)
tanh2radic120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radic120573 (1612057221205832+ 1)
1612057221205832+ 1
119905)
minus
31205732
16 (120573 + 1)
coth2radic120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radic120573 (1612057221205832+ 1)
1612057221205832+ 1
119905)
1199081(119909 119905) =
∓3radicminus41198814minus 41198812+ 21205732+ 4120573
8
plusmn
3radicminus41198814minus 41198812+ 21205732+ 4120573
16
tanh2radic120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radic120573 (1612057221205832+ 1)
1612057221205832+ 1
119905)
plusmn
3radicminus41198814minus 41198812+ 21205732+ 4120573
16
coth2radic120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radic120573 (1612057221205832+ 1)
1612057221205832+ 1
119905)
Journal of Applied Mathematics 9
1199062(119909 119905) =
120573 (120573 + 8)
8 (120573 + 1)
+
31205732
16 (120573 + 1)
tanh2radicminus120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radicminus120573 (1612057221205832minus 1)
1612057221205832minus 1
119905)
+
31205732
16 (120573 + 1)
coth2radicminus120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radicminus120573 (1612057221205832minus 1)
1612057221205832minus 1
119905)
1199082(119909 119905) = plusmn
radicminus41198814minus 41198812+ 21205732+ 4120573
8
plusmn
3radicminus41198814minus 41198812+ 21205732+ 4120573
16
tanh2radicminus120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radicminus120573 (1612057221205832minus 1)
1612057221205832minus 1
119905)
plusmn
3radicminus41198814minus 41198812+ 21205732+ 4120573
16
coth2radicminus120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radicminus120573 (1612057221205832minus 1)
1612057221205832minus 1
119905)
1199063(119909 119905) =
minus120573 (120573 minus 4)
4 (120573 + 1)
+
31205732
4 (120573 + 1)
tanh2radicminus120573 (120573 + 2)
2120572 (120573 + 2)
times (119909 minus
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
119905)
1199083(119909 119905) = ∓
radicminus41198814minus 41198812+ 21205732+ 4120573
4
plusmn
3radicminus41198814minus 41198812+ 21205732+ 4120573
4
tanhradicminus120573 (120573 + 2)
2120572 (120573 + 2)
times (119909 minus
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
119905)
1199064(119909 119905) =
120573 (3120573 + 4)
4 (120573 + 1)
minus
31205732
4 (120573 + 1)
tanh2radic120573 (120573 + 2)
2120572 (120573 + 2)
times (119909 minus
radic120573 (412057221205832+ 1)
412057221205832+ 1
119905)
1199084(119909 119905) = ∓
3radicminus41198814minus 41198812+ 21205732+ 4120573
4
plusmn
3radicminus41198814minus 41198812+ 21205732+ 4120573
4
tanh2radic120573 (120573 + 2)
2120572 (120573 + 2)
times (119909 minus
radic120573 (412057221205832+ 1)
412057221205832+ 1
119905)
1199065(119909 119905) =
120573
120573 + 1
+
31205732
2 (31205732+ 7120573 + 4)
tanh2radicminus120573 (5120573 + 4)
2120572 (5120573 + 4)
times (119909 minus
radic5120573 (812057221205832+ 3120573 + 5)
812057221205832+ 3120573 + 5
119905)
+
31205732
2 (31205732+ 7120573 + 4)
coth2radicminus120573 (5120573 + 4)
2120572 (5120573 + 4)
times (119909 minus
radic5120573 (812057221205832+ 3120573 + 5)
812057221205832+ 3120573 + 5
119905)
1199085(119909 119905)
= ∓
3radicminus421198814120573 minus 58119881
4minus 21198812120573 minus 120119881
2+ 120120573
8
times tanh2radicminus120573 (5120573 + 4)
2120572 (5120573 + 4)
times (119909 minus
radic5120573 (812057221205832+ 3120573 + 5)
812057221205832+ 3120573 + 5
119905)
plusmn
3radicminus421198814120573 minus 58119881
4minus 21198812120573 minus 120119881
2+ 120120573
8
10 Journal of Applied Mathematics
times coth2radicminus120573 (5120573 + 4)
2120572 (5120573 + 4)
times (119909 minus
radic5120573 (812057221205832+ 3120573 + 5)
812057221205832+ 3120573 + 5
119905)
1199066(119909 119905) =
120573
120573 + 1
minus
120573
2 (120573 + 1)
tanh2minus12120572
(119909 minus
radic3120573 (120573 + 1)
3 (120573 + 1)
119905)
minus
120573
2 (120573 + 1)
coth2minus12120572
(119909 minus
radic3120573 (120573 + 1)
3 (120573 + 1)
119905)
1199086(119909 119905)
= ∓
radic91198812+ 6
2
tanh2minus12120572
(119909 minus
radic3120573 (120573 + 1)
3 (120573 + 1)
119905)
plusmn
radic91198812+ 6
2
coth2minus12120572
(119909 minus
radic3120573 (120573 + 1)
3 (120573 + 1)
119905)
(33)
34 The Fourth Coupled We now consider the coupled IBqequations
119906119905119905= 119906119909119909minus 120573(119906
2)119909119909+ (1199082)119909119909+ 119906119909119909119905119905
119908119905119905= 119908119909119909+ 2(119906119908)
119909119909+ 119908119909119909119905119905
(34)
Using the wave variable 120585 = 119909 minus 119881119905 (34) can be converted toa system of ODEs
(1198812minus 1)119880 + 120573119880
2minus1198822minus 119881211988010158401015840= 0
(1198812minus 1)119882 minus 2119880119882 minus 119881
211988210158401015840= 0
(35)
Balancing1198822 with 11988010158401015840 and 119880119882 with11988210158401015840 in (35) gives
2119873 = 119872 + 2
119872 + 119873 = 119873 + 2
(36)
so that
119872 = 119873 = 2 (37)
The tanh-coth method substitutes the finite expansions
119880(120583120585) = 119878 (119884) =
2
sum
119896=0
119886119896119884119896+
2
sum
119896=1
119887119896119884minus119896
119882 (120583120585) = 119878 (119884) =
2
sum
119896=0
119888119896119884119896+
2
sum
119896=1
119889119896119884119896
(38)
In (35) collecting the coefficients of each power of 119884 andsolving the resulting system we obtain
1198860=
minus21205832
161205832+ 1
1198862=
minus31205832
161205832+ 1
1198862= 1198872
1198880=
21198892
3
1198882= 1198892
1198892=
plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
119881 =
1
radic161205832+ 1
1198861= 1198871= 1198881= 1198891= 0
1198860=
minus61205832
161205832minus 1
1198862=
31205832
161205832minus 1
1198862= 1198872
1198880= minus2119889
2 119888
2= 1198892
1198892=
plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
119881 =
1
radicminus161205832+ 1
1198861= 1198871= 1198881= 1198891= 0
1198860=
1205832
41205832+ 1
1198862=
minus31205832
41205832+ 1
1198880=
minus1198882
3
1198882=
plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
119881 =
1
radic41205832+ 1
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860=
minus31205832
41205832minus 1
1198862=
31205832
41205832minus 1
1198880= minus1198882
1198882=
plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
119881 =
1
radicminus41205832+ 1
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860=
minus31205832
41205832minus 1
1198872=
31205832
41205832minus 1
1198880= minus1198892
1198892=
plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
119881 =
1
radicminus41205832+ 1
1198861= 1198862= 1198871= 1198881= 1198882= 1198891= 0
(39)
Journal of Applied Mathematics 11
where 120583 is left as a free parameter Consequently we obtainthe following travelling wave solutions
1199061(119909 119905) = minus
21205832
161205832+ 1
minus
31205832
161205832+ 1
tanh2120583(119909 minus 1
radic161205832+ 1
119905)
minus
31205832
161205832+ 1
coth2120583(119909 minus 1
radic161205832+ 1
119905)
1199081(119909 119905) = plusmn
(119881 + 1) (119881 minus 1)radic120573 + 2
8
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
tanh2120583
times(119909 minus
1
radic161205832+ 1
119905)
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
coth2120583
times(119909 minus
1
radic161205832+ 1
119905)
1199062(119909 119905) = minus
61205832
161205832minus 1
+
31205832
161205832minus 1
tanh2120583(119909 minus 1
radicminus161205832+ 1
119905)
+
31205832
161205832minus 1
coth2120583(119909 minus 1
radicminus161205832+ 1
119905)
1199082(119909 119905) = ∓
3 (119881 + 1) (119881 minus 1)radic120573 + 2
8
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
tanh2120583
times(119909 minus
1
radicminus161205832+ 1
119905)
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
coth2120583
times(119909 minus
1
radicminus161205832+ 1
119905)
1199063(119909 119905) =
1205832
41205832+ 1
minus
31205832
41205832+ 1
tanh2120583(119909 minus 1
radic41205832+ 1
119905)
1199083(119909 119905) = ∓
(119881 + 1) (119881 minus 1)radic120573 + 2
4
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
times tanh2120583(119909 minus 1
radic41205832+ 1
119905)
1199064(119909 119905) = minus
31205832
41205832minus 1
+
31205832
41205832minus 1
tanh2120583(119909 minus 1
radicminus41205832+ 1
119905)
1199084(119909 119905) = ∓
3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
times tanh2120583(119909 minus 1
radicminus41205832+ 1
119905)
1199065(119909 119905) = minus
31205832
41205832minus 1
+
31205832
41205832minus 1
coth2120583(119909 minus 1
radicminus41205832+ 1
119905)
1199085(119909 119905) = ∓
3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
plusmn 3
(119881 + 1) (119881 minus 1)radic120573 + 2
4
times coth2120583(119909 minus 1
radicminus41205832+ 1
119905)
(40)
35 The Fifth Coupled We finally consider the coupled IBqequations
119906119905119905= 120573119906119909119909minus
120573
2
(1199062)119909119909+
120573
2
(1199082)119909119909+ 1205722119906119909119909119905119905
119908119905119905= 120574(119906119908)
119909119909+ 120588(119908
2)119909119909+ 1205722119908119909119909119905119905
(41)
12 Journal of Applied Mathematics
This system of equations (41) can be converted to the follow-ing system of ODEs by using the wave variable 120585 = 119909 minus 119881119905
(21198812minus 2120573)119880 + 120573119880
2minus 120573119882
2minus 21205722119881211988010158401015840= 0
1198812119882minus 120574119880119882 minus 120588119882
2minus 1205722119881211988210158401015840= 0
(42)
Using the balancing procedure in (42) gives
2119872 = 119872 +119873
2119873 = 119872 +119873
(43)
so that
119872 = 119873 = 2 (44)
We consider solutions in the following form
119880 (120583120585) = 119878 (119884) =
2
sum
119896=0
119886119896119884119896+
2
sum
119896=1
119887119896119884minus119896
119882 (120583120585) = 119878 (119884) =
2
sum
119896=0
119888119896119884119896+
2
sum
119896=1
119889119896119884119896
(45)
Substituting (45) into (42) collecting the coefficients of 119884and solving the resulting system we find the following setsof solutions
1198860= (minus120573 (8120588119888
21205732120574 minus 8120573120574
21205881198882minus 18120588
21205732+ 24120573120588
31198882
+1212057321205742minus 16120574
31205881198882+ 24120588
31198882120574 minus 3120574120573
3))
times (4 (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882+ 312058821205733
+ 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882minus 312057331205742minus 312057431205732))
minus1
1198862=
minus (41205881198882120573 + 4119888
2120588120574 minus 3120573
2)
4120574 (120583 + 120573)
1198880= (120573(120588
212057321198882+ 120574211988821205732+ 12057311988821205743+ 2120574212058821198882
+312057412058821205731198882minus
3
2
1205881205732120574 minus
3
4
1205881205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882+ 312058821205733
+ 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882minus 312057331205742minus 312057431205732)
minus1
1198882= (plusmn3 (minus 120573120588 + (minus120588
21205732+ 41205743120573 minus 4120574
21198814
minus 412057431198812+ 212057321205742
minus41205882120574120573 + 4120588
21198814+ 412058821198812120574)
12
))
times (4 (1205742minus 1205882))
minus1
119881 =
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
120583 =
radicminus120573 (2120574 + 120573)
2120572 (2120574 + 120573)
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860= (120573(6120588119888
21205732120574 + 10120573120574
21205881198882+
3
2
12058821205732+ 2120573120588
31198882
+312057321205742+ 412057431205881198882+ 212058831198882120574 +
9
4
1205741205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882minus 312058821205733
minus 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882+ 312057331205742+ 312057431205732)
minus1
1198862=
minus (41205881198882120573 + 4119888
2120588120574 + 3120573
2)
4120574 (120583 + 120573)
1198880= (120573(minus 3120588
212057321198882minus 3120574211988821205732minus 312057311988821205743
minus6120574212058821198882minus 9120574120588
21205731198882minus
9
2
1205881205732120574 minus
9
4
1205881205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882
minus 312058821205733minus 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882+ 312057331205742+ 312057431205732)
minus1
1198882= (plusmn3 (120573120588 + (minus120588
21205732+ 41205743120573 minus 4120574
21198814
minus 412057431198812+ 212057321205742minus 41205882120574120573
+412058821198814+ 412058821198812120574)
12
))
times (4 (1205742minus 1205882))
minus1
119881 =
radic120573 (412057221205832+ 1)
412057221205832+ 1
120583 =
radic120573 (2120574 + 120573)
2120572 (2120574 + 120573)
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
(46)
Journal of Applied Mathematics 13
In view of this we obtain the following travelling wavesolutions
1199061(119909 119905) = (minus120573 (8120588119888
21205732120574 minus 8120573120574
21205881198882minus 18120588
21205732+ 24120573120588
31198882
+1212057321205742minus 16120574
31205881198882+ 24120588
31198882120574 minus 3120574120573
3))
times (4 (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882+ 312058821205733
+ 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882minus 312057331205742minus 312057431205732))
minus1
minus
(41205881198882120573 + 4119888
2120588120574 minus 3120573
2)
4120574 (120583 + 120573)
tanh2radicminus120573 (2120574 + 120573)
2120572 (2120574 + 120573)
times(119909 minus
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
119905)
1199081(119909 119905) = (120573 (120588
212057321198882+ 120574211988821205732+ 12057311988821205743+ 2120574212058821198882
+312057412058821205731198882minus
3
2
1205881205732120574 minus
3
4
1205881205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882+ 312058821205733
+ 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882minus 312057331205742minus 312057431205732)
minus1
plusmn (3 (minus 120573120588 + (minus12058821205732+ 41205743120573 minus 4120574
21198814
minus 412057431198812+ 212057321205742minus 41205882120574120573
+412058821198814+ 412058821198812120574)
12
))
times (4 (1205742minus 1205882))
minus1
times tanh2radicminus120573 (2120574 + 120573)
2120572 (2120574 + 120573)
times(119909 minus
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
119905)
1199062(119909 119905) = (120573 (6120588119888
21205732120574 + 10120573120574
21205881198882+
3
2
12058821205732+ 2120573120588
31198882
+312057321205742+ 412057431205881198882+ 212058831198882120574 +
9
4
1205741205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882minus 312058821205733
minus 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882+ 312057331205742+ 312057431205732)
minus1
minus
(41205881198882120573 + 4119888
2120588120574 + 3120573
2)
4120574 (120583 + 120573)
tanh2radic120573 (2120574 + 120573)
2120572 (2120574 + 120573)
times(119909 minus
radic120573 (412057221205832+ 1)
412057221205832+ 1
119905)
1199082(119909 119905) = (120573 (minus 3120588
212057321198882minus 3120574211988821205732minus 312057311988821205743
minus6120574212058821198882minus 9120574120588
21205731198882minus
9
2
1205881205732120574 minus
9
4
1205881205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882minus 312058821205733
minus 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882+ 312057331205742+ 312057431205732)
minus1
plusmn (3 (120573120588 + (minus12058821205732+ 41205743120573 minus 4120574
21198814
minus 412057431198812+ 212057321205742minus 41205882120574120573
+412058821198814+ 412058821198812120574)
12
))
times (4 (1205742minus 1205882))
minus1
times tanh2radic120573 (2120574 + 120573)
2120572 (2120574 + 120573)
(119909 minus
radic120573 (412057221205832+ 1)
412057221205832+ 1
119905)
(47)
4 Conclusion
In summary the tanh-coth method has been successfullyimplemented to find new travelling wave solutions for severalkinds of two coupled improved Boussinesq equations Thesolutions that have been found can be seen as an expansionand verification of the previously known data However thisstudy confirmed again the belief that the tanh-coth methodis a powerful and reliable technique to handle nonlineardispersive-dissipative equations and system of equationsThroughout the work Maple has been used to overcome thetedious algebraic calculations
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] A De Godefroy ldquoBlow up of solutions of a generalized Boussi-nesq equationrdquo IMA Journal of AppliedMathematics vol 60 no2 pp 123ndash138 1998
14 Journal of Applied Mathematics
[2] S Wang and M Li ldquoThe Cauchy problem for coupled IMBqequationsrdquo IMA Journal of Applied Mathematics vol 74 no 5pp 726ndash740 2009
[3] N Duruk H A Erbay and A Erkip ldquoBlow-up and globalexistence for a general class of nonlocal nonlinear coupled waveequationsrdquo Journal of Differential Equations vol 250 no 3 pp1448ndash1459 2011
[4] G Chen and H Zhang ldquoInitial boundary value problem for asystem of generalized IMBq equationsrdquoMathematical Methodsin the Applied Sciences vol 27 no 5 pp 497ndash518 2004
[5] P Rosenau ldquoDynamics of dense latticesrdquo Physical Review B vol36 no 11 pp 5868ndash5876 1987
[6] P L Christiansen P S Lomdahl andVMuto ldquoOn aToda latticemodel with a transversal degree of freedomrdquo Nonlinearity vol4 no 2 pp 477ndash501 1991
[7] S K Turitsyn ldquoOn a Toda lattice model with a transversaldegree of freedom Sufficient criterion of blow-up in thecontinuum limitrdquo Physics Letters A vol 173 no 3 pp 267ndash2691993
[8] R L Pego P Smereka and M I Weinstein ldquoOscillatoryinstability of solitary waves in a continuum model of latticevibrationsrdquo Nonlinearity vol 8 no 6 pp 921ndash941 1995
[9] X Fan and L Tian ldquoCompacton solutions and multiple com-pacton solutions for a continuum Toda lattice modelrdquo ChaosSolitons and Fractals vol 29 no 4 pp 882ndash894 2006
[10] A M Wazwaz ldquoThe Hirotarsquos bilinear method and the tanh-coth method for multiple-soliton solutions of the Sawada-Kotera-Kadomtsev-Petviashvili equationrdquoAppliedMathematicsand Computation vol 200 no 1 pp 160ndash166 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 5
1199084(119909 119905) =
minus3120573 + 4120572
8120572 (120572 minus 120573)
+
3120573
8120572 (120572 minus 120573)
tanh2radic120573 (9120573 minus 12120572)
minus3120573 + 4120572
times(119909 minus
6
radic61205832+ 18
119905)
(20)
32 The Second Coupled We next consider the system ofequations
120588
120572
119906119905119905= 120573119906119909119909minus
1205732
2
(1199062)119909119909+
120573
2
(1199082)119909119909+
120588
120572
1198972
12
119906119909119909119905119905
120588
120572
119908119905119905= 120573(119906119908)
119909119909+
120588
120572
1198972
12
119908119909119909119905119905
(21)
Using thewave variable 120585 = 119909minus119881119905 and then by integrating thisequation twice and considering the constants of integration tobe zero the system (21) is carried to a system of ODEs
12 (120572120573 minus 1205881198812)119880 minus 6120572120573
21198802+ 6120572120573119882
2+ 1205881198972119881211988010158401015840= 0
12120572120573119880119882 + 1205881198972119881211988210158401015840minus 12120588119881
2119882 = 0
(22)
Balancing minus612057212057321198802 with 120588119897
2119881211988010158401015840 and 12120572120573119880119882 with
1205881198972119881211988210158401015840 in (22) gives
2119872 = 119872 + 2
2119873 = 119872 +119873
(23)
so that119872 = 119873 = 2 The tanh-coth method admits the use of
119880 (120583120585) = 119878 (119884) =
2
sum
119896=0
119886119896119884119896+
2
sum
119896=1
119887119896119884minus119896
119882 (120583120585) = 119878 (119884) =
2
sum
119896=0
119888119896119884119896+
2
sum
119896=1
119889119896119884119896
(24)
Substituting (24) into (22) collecting the coefficients of 119884and solving the resulting system we find the following setsof solutions
1198860=
1
120573 + 1
1198862=
minus1
2 (120573 + 1)
1198872= 1198862
1198892= plusmn
radic120573 + 2
2 (120573 + 1)
1198882= minus1198892
119881 =
radic3120588120572120573 (120573 + 1)
3120588 (120573 + 1)
120583 =
radic3
119897
1198861= 1198871= 1198880= 1198881= 1198891= 0
1198860=
1
120573 + 1
1198862=
3120573
61205732+ 14120573 + 8
1198872= 1198862
1198892= plusmn
3120573radic120573 + 2
2 (3120573 + 4) (120573 + 1)
1198882= minus1198892
119881 =
radic15120588120572120573 (211989721205832+ 9120573 + 15)
120588 (211989721205832+ 9120573 + 15)
120583 =
radicminus120573 (15120573 + 12)
(5120573 + 4) 119897
1198861= 1198871= 1198880= 1198881= 1198891= 0
1198860=
4 minus 120573
4 (120573 + 1)
1198862=
3120573
4120573 + 1
1198880=
minus1198882
3
1198882= plusmn
9120573
2radic611989721205832+ 36120573 + 18
119881 =
radicminus3120588120572120573 (minus3 + 11989721205832)
120588 (minus3 + 11989721205832)
120583 =
radicminus120573 (3120573 + 6)
(120573 + 2) 119897
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860=
3120573 + 4
4 (120573 + 1)
1198862=
minus3120573
4120573 + 1
1198880= minus1198882
1198882= plusmn
9120573
2radicminus611989721205832+ 36120573 + 18
119881 =
radic3120588120572120573 (3 + 11989721205832)
120588 (3 + 11989721205832)
120583 =
radic3120573 (120573 + 2)
(120573 + 2) 119897
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860=
120573 + 8
8 (120573 + 1)
1198862=
3120573
16 (120573 + 1)
1198872= 1198862 119888
0=
21198892
3
1198882= 1198892 119889
2= plusmn
9120573
8radic2411989721205832+ 36120573 + 18
119881 =
radicminus3120588120572120573 (minus3 + 411989721205832)
120588 (minus3 + 411989721205832)
120583 =
radicminus120573 (3120573 + 6)
2 (120573 + 2) 119897
1198861= 1198871= 1198881= 1198891= 0
1198860=
3 (120573 + 8)
8 (120573 + 1)
1198862=
minus3120573
16 (120573 + 1)
1198872= 1198862 119888
0= minus2119889
2
6 Journal of Applied Mathematics
1198882= 1198892 119889
2= plusmn
9120573
8radicminus2411989721205832+ 36120573 + 18
119881 =
radic3120588120572120573 (3 + 411989721205832)
120588 (3 + 411989721205832)
120583 =
radic120573 (3120573 + 6)
2 (120573 + 2) 119897
1198861= 1198871= 1198881= 1198891= 0
(25)
In accordance with these sets of solutions we have thefollowing solutions
1199061(119909 119905) =
1
120573 + 1
minus
1
2 (120573 + 1)
tanh2radic3
119897
times (119909 minus
radic3120588120572120573 (120573 + 1)
3120588 (120573 + 1)
119905)
minus
1
2 (120573 + 1)
coth2radic3
119897
(119909 minus
radic3120588120572120573 (120573 + 1)
3120588 (120573 + 1)
119905)
1199081(119909 119905) = ∓
radic120573 + 2
2 (120573 + 1)
tanh2radic3
119897
(119909 minus
radic3120588120572120573 (120573 + 1)
3120588 (120573 + 1)
119905)
plusmn
radic120573 + 2
2 (120573 + 1)
coth2radic3
119897
(119909 minus
radic3120588120572120573 (120573 + 1)
3120588 (120573 + 1)
119905)
1199062(119909 119905) =
1
120573 + 1
+
3120573
61205732+ 14120573 + 8
tanh2radicminus120573 (15120573 + 12)
(5120573 + 4) 119897
times (119909 minus
radic15120588120572120573 (211989721205832+ 9120573 + 15)
120588 (211989721205832+ 9120573 + 15)
119905)
+
3120573
61205732+ 14120573 + 8
coth2radicminus120573 (15120573 + 12)
(5120573 + 4) 119897
times (119909 minus
radic15120588120572120573 (211989721205832+ 9120573 + 15)
120588 (211989721205832+ 9120573 + 15)
119905)
1199082(119909 119905) = ∓
3120573radic120573 + 2
2 (3120573 + 4) (120573 + 1)
tanh2radicminus120573 (15120573 + 12)
(5120573 + 4) 119897
times (119909 minus
radic15120588120572120573 (211989721205832+ 9120573 + 15)
120588 (211989721205832+ 9120573 + 15)
119905)
plusmn
3120573radic120573 + 2
2 (3120573 + 4) (120573 + 1)
coth2radicminus120573 (15120573 + 12)
(5120573 + 4) 119897
times (119909 minus
radic15120588120572120573 (211989721205832+ 9120573 + 15)
120588 (211989721205832+ 9120573 + 15)
119905)
1199063(119909 119905) =
4 minus 120573
4 (120573 + 1)
+
3120573
4120573 + 1
tanh2radicminus120573 (3120573 + 6)
(120573 + 2) 119897
times (119909 minus
radicminus3120588120572120573 (minus3 + 11989721205832)
120588 (minus3 + 11989721205832)
119905)
1199083(119909 119905) = ∓
3120573
2radic611989721205832+ 36120573 + 18
plusmn
9120573
2radic611989721205832+ 36120573 + 18
tanh2radicminus120573 (3120573 + 6)
(120573 + 2) 119897
times (119909 minus
radicminus3120588120572120573 (minus3 + 11989721205832)
120588 (minus3 + 11989721205832)
119905)
1199064(119909 119905) =
3120573 + 4
4 (120573 + 1)
minus
3120573
4120573 + 1
tanh2radic3120573 (120573 + 2)
(120573 + 2) 119897
times (119909 minus
radic3120588120572120573 (3 + 11989721205832)
120588 (3 + 11989721205832)
119905)
1199084(119909 119905) = ∓
9120573
2radicminus611989721205832+ 36120573 + 18
plusmn
9120573
2radicminus611989721205832+ 36120573 + 18
tanh2radic3120573 (120573 + 2)
(120573 + 2) 119897
times (119909 minus
radic3120588120572120573 (3 + 11989721205832)
120588 (3 + 11989721205832)
119905)
1199065(119909 119905) =
120573 + 8
8 (120573 + 1)
+
3120573
16 (120573 + 1)
tanh2radicminus120573 (3120573 + 6)
2 (120573 + 2) 119897
Journal of Applied Mathematics 7
times(119909 minus
radicminus3120588120572120573 (minus3 + 411989721205832)
120588 (minus3 + 411989721205832)
119905)
+
3120573
16 (120573 + 1)
coth2radicminus120573 (3120573 + 6)
2 (120573 + 2) 119897
times (119909 minus
radicminus3120588120572120573 (minus3 + 411989721205832)
120588 (minus3 + 411989721205832)
119905)
1199085(119909 119905) = plusmn
3120573
4radic2411989721205832+ 36120573 + 18
plusmn
9120573
8radic2411989721205832+ 36120573 + 18
tanh2radicminus120573 (3120573 + 6)
2 (120573 + 2) 119897
times (119909 minus
radicminus3120588120572120573 (minus3 + 411989721205832)
120588 (minus3 + 411989721205832)
119905)
plusmn
9120573
8radic2411989721205832+ 36120573 + 18
coth2radicminus120573 (3120573 + 6)
2 (120573 + 2) 119897
times (119909 minus
radicminus3120588120572120573 (minus3 + 411989721205832)
120588 (minus3 + 411989721205832)
119905)
1199066(119909 119905) =
3 (120573 + 8)
8 (120573 + 1)
minus
3120573
16 (120573 + 1)
tanh2radic120573 (3120573 + 6)
2 (120573 + 2) 119897
times (119909 minus
radic3120588120572120573 (3 + 411989721205832)
120588 (3 + 411989721205832)
119905)
minus
3120573
16 (120573 + 1)
coth2radic120573 (3120573 + 6)
2 (120573 + 2) 119897
times (119909 minus
radic3120588120572120573 (3 + 411989721205832)
120588 (3 + 411989721205832)
119905)
1199086(119909 119905) = ∓
9120573
4radicminus2411989721205832+ 36120573 + 18
plusmn
9120573
8radicminus2411989721205832+ 36120573 + 18
tanh2radic120573 (3120573 + 6)
2 (120573 + 2) 119897
times (119909 minus
radic3120588120572120573 (3 + 411989721205832)
120588 (3 + 411989721205832)
119905)
plusmn
9120573
8radicminus2411989721205832+ 36120573 + 18
coth2radic120573 (3120573 + 6)
2 (120573 + 2) 119897
times (119909 minus
radic3120588120572120573 (3 + 411989721205832)
120588 (3 + 411989721205832)
119905)
(26)
33 The Third Coupled We now examine the coupled equa-tions
119906119905119905= 120573119906119909119909minus
120573
2
(1199062)119909119909+
1
2
(1199082)119909119909+ 1205722119906119909119909119905119905
119908119905119905= (119906119908)
119909119909+ 1205722119908119909119909119905119905
(27)
Using the wave variable 120585 = 119909 minus119881119905 the system (27) is carriedto a system of ODEs Integrating these equations twice andneglecting constants of integration we find
2 (1198812minus 120573)119880 + 120573119880
2minus 120573119882
2minus 21205722119881211988010158401015840= 0
1198812119882minus119880119882 minus 120572
2119881211988210158401015840= 0
(28)
Balancing 1205731198822 with 21205722119881211988010158401015840 and 119881211988210158401015840 with 119880119882 in (28)gives
2119873 = 119872 + 2
2 + 119873 = 119872 +119873
(29)
so that
119872 = 119873 = 2 (30)
As a result we seek solutions to (27) in the form
119880(120583120585) = 119878 (119884) =
2
sum
119896=0
119886119896119884119896+
2
sum
119896=1
119887119896119884minus119896
119882 (120583120585) = 119878 (119884) =
2
sum
119896=0
119888119896119884119896+
2
sum
119896=1
119889119896119884119896
(31)
Substitution of (31) into (28) leads to an algebraic system inthe unknowns 119886
0 1198861 1198862 1198871 1198872 1198880 1198881 1198882 1198891 1198892 119881 and 120583
8 Journal of Applied Mathematics
Collecting the coefficients of 119884 and solving the resultingsystem we find the following sets of solutions
1198860=
120573 (3120573 + 8)
8 (120573 + 1)
1198862=
minus31205732
16 (120573 + 1)
1198872= 1198862 119888
0= minus2119889
2 119888
2= 1198892
1198892=
plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573
16
119881 =
radic120573 (1612057221205832+ 1)
1612057221205832+ 1
120583 =
radic120573 (120573 + 2)
4120572 (120573 + 2)
1198861= 1198871= 1198881= 1198891= 0
1198860=
120573 (120573 + 8)
8 (120573 + 1)
1198862=
31205732
16 (120573 + 1)
1198872= 1198862
1198880=
21198892
3
1198882= 1198892
1198892=
plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573
16
119881 =
radicminus120573 (1612057221205832minus 1)
1612057221205832minus 1
120583 =
radicminus120573 (120573 + 2)
4120572 (120573 + 2)
1198861= 1198871= 1198881= 1198891= 0
1198860=
minus120573 (120573 minus 4)
4 (120573 + 1)
1198862=
31205732
4 (120573 + 1)
1198880= minus
1198882
3
1198882=
plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573
4
119881 =
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
120583 =
radicminus120573 (120573 + 2)
2120572 (120573 + 2)
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860=
120573 (3120573 + 4)
4 (120573 + 1)
1198862=
minus31205732
4 (120573 + 1)
1198880= minus2
1198882=
plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573
4
119881 =
radic120573 (412057221205832+ 1)
412057221205832+ 1
120583 =
radic120573 (120573 + 2)
2120572 (120573 + 2)
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860=
120573
120573 + 1
1198862=
31205732
2 (31205732+ 7120573 + 4)
1198872= 1198862 119888
2= minus1198892
1198892=
plusmn3radicminus421198814120573 minus 58119881
4minus 21198812120573 minus 120119881
2+ 120120573
8
119881 =
radic5120573 (812057221205832+ 3120573 + 5)
812057221205832+ 3120573 + 5
120583 =
radicminus120573 (5120573 + 4)
2120572 (5120573 + 4)
1198861= 1198871= 1198880= 1198881= 1198891= 0
1198860=
120573
120573 + 1
1198862=
minus120573
2 (120573 + 1)
1198872= 1198862
1198882= minus1198892 119889
2=
plusmnradic91198812+ 6
2
119881 =
radic3120573 (120573 + 1)
3 (120573 + 1)
120583 =
minus1
2120572
1198861= 1198871= 1198880= 1198881= 1198891= 0
(32)
Consequently we obtain the following solutions
1199061(119909 119905) =
120573 (3120573 + 8)
8 (120573 + 1)
minus
31205732
16 (120573 + 1)
tanh2radic120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radic120573 (1612057221205832+ 1)
1612057221205832+ 1
119905)
minus
31205732
16 (120573 + 1)
coth2radic120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radic120573 (1612057221205832+ 1)
1612057221205832+ 1
119905)
1199081(119909 119905) =
∓3radicminus41198814minus 41198812+ 21205732+ 4120573
8
plusmn
3radicminus41198814minus 41198812+ 21205732+ 4120573
16
tanh2radic120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radic120573 (1612057221205832+ 1)
1612057221205832+ 1
119905)
plusmn
3radicminus41198814minus 41198812+ 21205732+ 4120573
16
coth2radic120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radic120573 (1612057221205832+ 1)
1612057221205832+ 1
119905)
Journal of Applied Mathematics 9
1199062(119909 119905) =
120573 (120573 + 8)
8 (120573 + 1)
+
31205732
16 (120573 + 1)
tanh2radicminus120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radicminus120573 (1612057221205832minus 1)
1612057221205832minus 1
119905)
+
31205732
16 (120573 + 1)
coth2radicminus120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radicminus120573 (1612057221205832minus 1)
1612057221205832minus 1
119905)
1199082(119909 119905) = plusmn
radicminus41198814minus 41198812+ 21205732+ 4120573
8
plusmn
3radicminus41198814minus 41198812+ 21205732+ 4120573
16
tanh2radicminus120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radicminus120573 (1612057221205832minus 1)
1612057221205832minus 1
119905)
plusmn
3radicminus41198814minus 41198812+ 21205732+ 4120573
16
coth2radicminus120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radicminus120573 (1612057221205832minus 1)
1612057221205832minus 1
119905)
1199063(119909 119905) =
minus120573 (120573 minus 4)
4 (120573 + 1)
+
31205732
4 (120573 + 1)
tanh2radicminus120573 (120573 + 2)
2120572 (120573 + 2)
times (119909 minus
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
119905)
1199083(119909 119905) = ∓
radicminus41198814minus 41198812+ 21205732+ 4120573
4
plusmn
3radicminus41198814minus 41198812+ 21205732+ 4120573
4
tanhradicminus120573 (120573 + 2)
2120572 (120573 + 2)
times (119909 minus
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
119905)
1199064(119909 119905) =
120573 (3120573 + 4)
4 (120573 + 1)
minus
31205732
4 (120573 + 1)
tanh2radic120573 (120573 + 2)
2120572 (120573 + 2)
times (119909 minus
radic120573 (412057221205832+ 1)
412057221205832+ 1
119905)
1199084(119909 119905) = ∓
3radicminus41198814minus 41198812+ 21205732+ 4120573
4
plusmn
3radicminus41198814minus 41198812+ 21205732+ 4120573
4
tanh2radic120573 (120573 + 2)
2120572 (120573 + 2)
times (119909 minus
radic120573 (412057221205832+ 1)
412057221205832+ 1
119905)
1199065(119909 119905) =
120573
120573 + 1
+
31205732
2 (31205732+ 7120573 + 4)
tanh2radicminus120573 (5120573 + 4)
2120572 (5120573 + 4)
times (119909 minus
radic5120573 (812057221205832+ 3120573 + 5)
812057221205832+ 3120573 + 5
119905)
+
31205732
2 (31205732+ 7120573 + 4)
coth2radicminus120573 (5120573 + 4)
2120572 (5120573 + 4)
times (119909 minus
radic5120573 (812057221205832+ 3120573 + 5)
812057221205832+ 3120573 + 5
119905)
1199085(119909 119905)
= ∓
3radicminus421198814120573 minus 58119881
4minus 21198812120573 minus 120119881
2+ 120120573
8
times tanh2radicminus120573 (5120573 + 4)
2120572 (5120573 + 4)
times (119909 minus
radic5120573 (812057221205832+ 3120573 + 5)
812057221205832+ 3120573 + 5
119905)
plusmn
3radicminus421198814120573 minus 58119881
4minus 21198812120573 minus 120119881
2+ 120120573
8
10 Journal of Applied Mathematics
times coth2radicminus120573 (5120573 + 4)
2120572 (5120573 + 4)
times (119909 minus
radic5120573 (812057221205832+ 3120573 + 5)
812057221205832+ 3120573 + 5
119905)
1199066(119909 119905) =
120573
120573 + 1
minus
120573
2 (120573 + 1)
tanh2minus12120572
(119909 minus
radic3120573 (120573 + 1)
3 (120573 + 1)
119905)
minus
120573
2 (120573 + 1)
coth2minus12120572
(119909 minus
radic3120573 (120573 + 1)
3 (120573 + 1)
119905)
1199086(119909 119905)
= ∓
radic91198812+ 6
2
tanh2minus12120572
(119909 minus
radic3120573 (120573 + 1)
3 (120573 + 1)
119905)
plusmn
radic91198812+ 6
2
coth2minus12120572
(119909 minus
radic3120573 (120573 + 1)
3 (120573 + 1)
119905)
(33)
34 The Fourth Coupled We now consider the coupled IBqequations
119906119905119905= 119906119909119909minus 120573(119906
2)119909119909+ (1199082)119909119909+ 119906119909119909119905119905
119908119905119905= 119908119909119909+ 2(119906119908)
119909119909+ 119908119909119909119905119905
(34)
Using the wave variable 120585 = 119909 minus 119881119905 (34) can be converted toa system of ODEs
(1198812minus 1)119880 + 120573119880
2minus1198822minus 119881211988010158401015840= 0
(1198812minus 1)119882 minus 2119880119882 minus 119881
211988210158401015840= 0
(35)
Balancing1198822 with 11988010158401015840 and 119880119882 with11988210158401015840 in (35) gives
2119873 = 119872 + 2
119872 + 119873 = 119873 + 2
(36)
so that
119872 = 119873 = 2 (37)
The tanh-coth method substitutes the finite expansions
119880(120583120585) = 119878 (119884) =
2
sum
119896=0
119886119896119884119896+
2
sum
119896=1
119887119896119884minus119896
119882 (120583120585) = 119878 (119884) =
2
sum
119896=0
119888119896119884119896+
2
sum
119896=1
119889119896119884119896
(38)
In (35) collecting the coefficients of each power of 119884 andsolving the resulting system we obtain
1198860=
minus21205832
161205832+ 1
1198862=
minus31205832
161205832+ 1
1198862= 1198872
1198880=
21198892
3
1198882= 1198892
1198892=
plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
119881 =
1
radic161205832+ 1
1198861= 1198871= 1198881= 1198891= 0
1198860=
minus61205832
161205832minus 1
1198862=
31205832
161205832minus 1
1198862= 1198872
1198880= minus2119889
2 119888
2= 1198892
1198892=
plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
119881 =
1
radicminus161205832+ 1
1198861= 1198871= 1198881= 1198891= 0
1198860=
1205832
41205832+ 1
1198862=
minus31205832
41205832+ 1
1198880=
minus1198882
3
1198882=
plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
119881 =
1
radic41205832+ 1
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860=
minus31205832
41205832minus 1
1198862=
31205832
41205832minus 1
1198880= minus1198882
1198882=
plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
119881 =
1
radicminus41205832+ 1
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860=
minus31205832
41205832minus 1
1198872=
31205832
41205832minus 1
1198880= minus1198892
1198892=
plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
119881 =
1
radicminus41205832+ 1
1198861= 1198862= 1198871= 1198881= 1198882= 1198891= 0
(39)
Journal of Applied Mathematics 11
where 120583 is left as a free parameter Consequently we obtainthe following travelling wave solutions
1199061(119909 119905) = minus
21205832
161205832+ 1
minus
31205832
161205832+ 1
tanh2120583(119909 minus 1
radic161205832+ 1
119905)
minus
31205832
161205832+ 1
coth2120583(119909 minus 1
radic161205832+ 1
119905)
1199081(119909 119905) = plusmn
(119881 + 1) (119881 minus 1)radic120573 + 2
8
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
tanh2120583
times(119909 minus
1
radic161205832+ 1
119905)
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
coth2120583
times(119909 minus
1
radic161205832+ 1
119905)
1199062(119909 119905) = minus
61205832
161205832minus 1
+
31205832
161205832minus 1
tanh2120583(119909 minus 1
radicminus161205832+ 1
119905)
+
31205832
161205832minus 1
coth2120583(119909 minus 1
radicminus161205832+ 1
119905)
1199082(119909 119905) = ∓
3 (119881 + 1) (119881 minus 1)radic120573 + 2
8
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
tanh2120583
times(119909 minus
1
radicminus161205832+ 1
119905)
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
coth2120583
times(119909 minus
1
radicminus161205832+ 1
119905)
1199063(119909 119905) =
1205832
41205832+ 1
minus
31205832
41205832+ 1
tanh2120583(119909 minus 1
radic41205832+ 1
119905)
1199083(119909 119905) = ∓
(119881 + 1) (119881 minus 1)radic120573 + 2
4
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
times tanh2120583(119909 minus 1
radic41205832+ 1
119905)
1199064(119909 119905) = minus
31205832
41205832minus 1
+
31205832
41205832minus 1
tanh2120583(119909 minus 1
radicminus41205832+ 1
119905)
1199084(119909 119905) = ∓
3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
times tanh2120583(119909 minus 1
radicminus41205832+ 1
119905)
1199065(119909 119905) = minus
31205832
41205832minus 1
+
31205832
41205832minus 1
coth2120583(119909 minus 1
radicminus41205832+ 1
119905)
1199085(119909 119905) = ∓
3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
plusmn 3
(119881 + 1) (119881 minus 1)radic120573 + 2
4
times coth2120583(119909 minus 1
radicminus41205832+ 1
119905)
(40)
35 The Fifth Coupled We finally consider the coupled IBqequations
119906119905119905= 120573119906119909119909minus
120573
2
(1199062)119909119909+
120573
2
(1199082)119909119909+ 1205722119906119909119909119905119905
119908119905119905= 120574(119906119908)
119909119909+ 120588(119908
2)119909119909+ 1205722119908119909119909119905119905
(41)
12 Journal of Applied Mathematics
This system of equations (41) can be converted to the follow-ing system of ODEs by using the wave variable 120585 = 119909 minus 119881119905
(21198812minus 2120573)119880 + 120573119880
2minus 120573119882
2minus 21205722119881211988010158401015840= 0
1198812119882minus 120574119880119882 minus 120588119882
2minus 1205722119881211988210158401015840= 0
(42)
Using the balancing procedure in (42) gives
2119872 = 119872 +119873
2119873 = 119872 +119873
(43)
so that
119872 = 119873 = 2 (44)
We consider solutions in the following form
119880 (120583120585) = 119878 (119884) =
2
sum
119896=0
119886119896119884119896+
2
sum
119896=1
119887119896119884minus119896
119882 (120583120585) = 119878 (119884) =
2
sum
119896=0
119888119896119884119896+
2
sum
119896=1
119889119896119884119896
(45)
Substituting (45) into (42) collecting the coefficients of 119884and solving the resulting system we find the following setsof solutions
1198860= (minus120573 (8120588119888
21205732120574 minus 8120573120574
21205881198882minus 18120588
21205732+ 24120573120588
31198882
+1212057321205742minus 16120574
31205881198882+ 24120588
31198882120574 minus 3120574120573
3))
times (4 (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882+ 312058821205733
+ 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882minus 312057331205742minus 312057431205732))
minus1
1198862=
minus (41205881198882120573 + 4119888
2120588120574 minus 3120573
2)
4120574 (120583 + 120573)
1198880= (120573(120588
212057321198882+ 120574211988821205732+ 12057311988821205743+ 2120574212058821198882
+312057412058821205731198882minus
3
2
1205881205732120574 minus
3
4
1205881205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882+ 312058821205733
+ 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882minus 312057331205742minus 312057431205732)
minus1
1198882= (plusmn3 (minus 120573120588 + (minus120588
21205732+ 41205743120573 minus 4120574
21198814
minus 412057431198812+ 212057321205742
minus41205882120574120573 + 4120588
21198814+ 412058821198812120574)
12
))
times (4 (1205742minus 1205882))
minus1
119881 =
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
120583 =
radicminus120573 (2120574 + 120573)
2120572 (2120574 + 120573)
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860= (120573(6120588119888
21205732120574 + 10120573120574
21205881198882+
3
2
12058821205732+ 2120573120588
31198882
+312057321205742+ 412057431205881198882+ 212058831198882120574 +
9
4
1205741205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882minus 312058821205733
minus 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882+ 312057331205742+ 312057431205732)
minus1
1198862=
minus (41205881198882120573 + 4119888
2120588120574 + 3120573
2)
4120574 (120583 + 120573)
1198880= (120573(minus 3120588
212057321198882minus 3120574211988821205732minus 312057311988821205743
minus6120574212058821198882minus 9120574120588
21205731198882minus
9
2
1205881205732120574 minus
9
4
1205881205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882
minus 312058821205733minus 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882+ 312057331205742+ 312057431205732)
minus1
1198882= (plusmn3 (120573120588 + (minus120588
21205732+ 41205743120573 minus 4120574
21198814
minus 412057431198812+ 212057321205742minus 41205882120574120573
+412058821198814+ 412058821198812120574)
12
))
times (4 (1205742minus 1205882))
minus1
119881 =
radic120573 (412057221205832+ 1)
412057221205832+ 1
120583 =
radic120573 (2120574 + 120573)
2120572 (2120574 + 120573)
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
(46)
Journal of Applied Mathematics 13
In view of this we obtain the following travelling wavesolutions
1199061(119909 119905) = (minus120573 (8120588119888
21205732120574 minus 8120573120574
21205881198882minus 18120588
21205732+ 24120573120588
31198882
+1212057321205742minus 16120574
31205881198882+ 24120588
31198882120574 minus 3120574120573
3))
times (4 (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882+ 312058821205733
+ 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882minus 312057331205742minus 312057431205732))
minus1
minus
(41205881198882120573 + 4119888
2120588120574 minus 3120573
2)
4120574 (120583 + 120573)
tanh2radicminus120573 (2120574 + 120573)
2120572 (2120574 + 120573)
times(119909 minus
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
119905)
1199081(119909 119905) = (120573 (120588
212057321198882+ 120574211988821205732+ 12057311988821205743+ 2120574212058821198882
+312057412058821205731198882minus
3
2
1205881205732120574 minus
3
4
1205881205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882+ 312058821205733
+ 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882minus 312057331205742minus 312057431205732)
minus1
plusmn (3 (minus 120573120588 + (minus12058821205732+ 41205743120573 minus 4120574
21198814
minus 412057431198812+ 212057321205742minus 41205882120574120573
+412058821198814+ 412058821198812120574)
12
))
times (4 (1205742minus 1205882))
minus1
times tanh2radicminus120573 (2120574 + 120573)
2120572 (2120574 + 120573)
times(119909 minus
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
119905)
1199062(119909 119905) = (120573 (6120588119888
21205732120574 + 10120573120574
21205881198882+
3
2
12058821205732+ 2120573120588
31198882
+312057321205742+ 412057431205881198882+ 212058831198882120574 +
9
4
1205741205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882minus 312058821205733
minus 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882+ 312057331205742+ 312057431205732)
minus1
minus
(41205881198882120573 + 4119888
2120588120574 + 3120573
2)
4120574 (120583 + 120573)
tanh2radic120573 (2120574 + 120573)
2120572 (2120574 + 120573)
times(119909 minus
radic120573 (412057221205832+ 1)
412057221205832+ 1
119905)
1199082(119909 119905) = (120573 (minus 3120588
212057321198882minus 3120574211988821205732minus 312057311988821205743
minus6120574212058821198882minus 9120574120588
21205731198882minus
9
2
1205881205732120574 minus
9
4
1205881205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882minus 312058821205733
minus 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882+ 312057331205742+ 312057431205732)
minus1
plusmn (3 (120573120588 + (minus12058821205732+ 41205743120573 minus 4120574
21198814
minus 412057431198812+ 212057321205742minus 41205882120574120573
+412058821198814+ 412058821198812120574)
12
))
times (4 (1205742minus 1205882))
minus1
times tanh2radic120573 (2120574 + 120573)
2120572 (2120574 + 120573)
(119909 minus
radic120573 (412057221205832+ 1)
412057221205832+ 1
119905)
(47)
4 Conclusion
In summary the tanh-coth method has been successfullyimplemented to find new travelling wave solutions for severalkinds of two coupled improved Boussinesq equations Thesolutions that have been found can be seen as an expansionand verification of the previously known data However thisstudy confirmed again the belief that the tanh-coth methodis a powerful and reliable technique to handle nonlineardispersive-dissipative equations and system of equationsThroughout the work Maple has been used to overcome thetedious algebraic calculations
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] A De Godefroy ldquoBlow up of solutions of a generalized Boussi-nesq equationrdquo IMA Journal of AppliedMathematics vol 60 no2 pp 123ndash138 1998
14 Journal of Applied Mathematics
[2] S Wang and M Li ldquoThe Cauchy problem for coupled IMBqequationsrdquo IMA Journal of Applied Mathematics vol 74 no 5pp 726ndash740 2009
[3] N Duruk H A Erbay and A Erkip ldquoBlow-up and globalexistence for a general class of nonlocal nonlinear coupled waveequationsrdquo Journal of Differential Equations vol 250 no 3 pp1448ndash1459 2011
[4] G Chen and H Zhang ldquoInitial boundary value problem for asystem of generalized IMBq equationsrdquoMathematical Methodsin the Applied Sciences vol 27 no 5 pp 497ndash518 2004
[5] P Rosenau ldquoDynamics of dense latticesrdquo Physical Review B vol36 no 11 pp 5868ndash5876 1987
[6] P L Christiansen P S Lomdahl andVMuto ldquoOn aToda latticemodel with a transversal degree of freedomrdquo Nonlinearity vol4 no 2 pp 477ndash501 1991
[7] S K Turitsyn ldquoOn a Toda lattice model with a transversaldegree of freedom Sufficient criterion of blow-up in thecontinuum limitrdquo Physics Letters A vol 173 no 3 pp 267ndash2691993
[8] R L Pego P Smereka and M I Weinstein ldquoOscillatoryinstability of solitary waves in a continuum model of latticevibrationsrdquo Nonlinearity vol 8 no 6 pp 921ndash941 1995
[9] X Fan and L Tian ldquoCompacton solutions and multiple com-pacton solutions for a continuum Toda lattice modelrdquo ChaosSolitons and Fractals vol 29 no 4 pp 882ndash894 2006
[10] A M Wazwaz ldquoThe Hirotarsquos bilinear method and the tanh-coth method for multiple-soliton solutions of the Sawada-Kotera-Kadomtsev-Petviashvili equationrdquoAppliedMathematicsand Computation vol 200 no 1 pp 160ndash166 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Journal of Applied Mathematics
1198882= 1198892 119889
2= plusmn
9120573
8radicminus2411989721205832+ 36120573 + 18
119881 =
radic3120588120572120573 (3 + 411989721205832)
120588 (3 + 411989721205832)
120583 =
radic120573 (3120573 + 6)
2 (120573 + 2) 119897
1198861= 1198871= 1198881= 1198891= 0
(25)
In accordance with these sets of solutions we have thefollowing solutions
1199061(119909 119905) =
1
120573 + 1
minus
1
2 (120573 + 1)
tanh2radic3
119897
times (119909 minus
radic3120588120572120573 (120573 + 1)
3120588 (120573 + 1)
119905)
minus
1
2 (120573 + 1)
coth2radic3
119897
(119909 minus
radic3120588120572120573 (120573 + 1)
3120588 (120573 + 1)
119905)
1199081(119909 119905) = ∓
radic120573 + 2
2 (120573 + 1)
tanh2radic3
119897
(119909 minus
radic3120588120572120573 (120573 + 1)
3120588 (120573 + 1)
119905)
plusmn
radic120573 + 2
2 (120573 + 1)
coth2radic3
119897
(119909 minus
radic3120588120572120573 (120573 + 1)
3120588 (120573 + 1)
119905)
1199062(119909 119905) =
1
120573 + 1
+
3120573
61205732+ 14120573 + 8
tanh2radicminus120573 (15120573 + 12)
(5120573 + 4) 119897
times (119909 minus
radic15120588120572120573 (211989721205832+ 9120573 + 15)
120588 (211989721205832+ 9120573 + 15)
119905)
+
3120573
61205732+ 14120573 + 8
coth2radicminus120573 (15120573 + 12)
(5120573 + 4) 119897
times (119909 minus
radic15120588120572120573 (211989721205832+ 9120573 + 15)
120588 (211989721205832+ 9120573 + 15)
119905)
1199082(119909 119905) = ∓
3120573radic120573 + 2
2 (3120573 + 4) (120573 + 1)
tanh2radicminus120573 (15120573 + 12)
(5120573 + 4) 119897
times (119909 minus
radic15120588120572120573 (211989721205832+ 9120573 + 15)
120588 (211989721205832+ 9120573 + 15)
119905)
plusmn
3120573radic120573 + 2
2 (3120573 + 4) (120573 + 1)
coth2radicminus120573 (15120573 + 12)
(5120573 + 4) 119897
times (119909 minus
radic15120588120572120573 (211989721205832+ 9120573 + 15)
120588 (211989721205832+ 9120573 + 15)
119905)
1199063(119909 119905) =
4 minus 120573
4 (120573 + 1)
+
3120573
4120573 + 1
tanh2radicminus120573 (3120573 + 6)
(120573 + 2) 119897
times (119909 minus
radicminus3120588120572120573 (minus3 + 11989721205832)
120588 (minus3 + 11989721205832)
119905)
1199083(119909 119905) = ∓
3120573
2radic611989721205832+ 36120573 + 18
plusmn
9120573
2radic611989721205832+ 36120573 + 18
tanh2radicminus120573 (3120573 + 6)
(120573 + 2) 119897
times (119909 minus
radicminus3120588120572120573 (minus3 + 11989721205832)
120588 (minus3 + 11989721205832)
119905)
1199064(119909 119905) =
3120573 + 4
4 (120573 + 1)
minus
3120573
4120573 + 1
tanh2radic3120573 (120573 + 2)
(120573 + 2) 119897
times (119909 minus
radic3120588120572120573 (3 + 11989721205832)
120588 (3 + 11989721205832)
119905)
1199084(119909 119905) = ∓
9120573
2radicminus611989721205832+ 36120573 + 18
plusmn
9120573
2radicminus611989721205832+ 36120573 + 18
tanh2radic3120573 (120573 + 2)
(120573 + 2) 119897
times (119909 minus
radic3120588120572120573 (3 + 11989721205832)
120588 (3 + 11989721205832)
119905)
1199065(119909 119905) =
120573 + 8
8 (120573 + 1)
+
3120573
16 (120573 + 1)
tanh2radicminus120573 (3120573 + 6)
2 (120573 + 2) 119897
Journal of Applied Mathematics 7
times(119909 minus
radicminus3120588120572120573 (minus3 + 411989721205832)
120588 (minus3 + 411989721205832)
119905)
+
3120573
16 (120573 + 1)
coth2radicminus120573 (3120573 + 6)
2 (120573 + 2) 119897
times (119909 minus
radicminus3120588120572120573 (minus3 + 411989721205832)
120588 (minus3 + 411989721205832)
119905)
1199085(119909 119905) = plusmn
3120573
4radic2411989721205832+ 36120573 + 18
plusmn
9120573
8radic2411989721205832+ 36120573 + 18
tanh2radicminus120573 (3120573 + 6)
2 (120573 + 2) 119897
times (119909 minus
radicminus3120588120572120573 (minus3 + 411989721205832)
120588 (minus3 + 411989721205832)
119905)
plusmn
9120573
8radic2411989721205832+ 36120573 + 18
coth2radicminus120573 (3120573 + 6)
2 (120573 + 2) 119897
times (119909 minus
radicminus3120588120572120573 (minus3 + 411989721205832)
120588 (minus3 + 411989721205832)
119905)
1199066(119909 119905) =
3 (120573 + 8)
8 (120573 + 1)
minus
3120573
16 (120573 + 1)
tanh2radic120573 (3120573 + 6)
2 (120573 + 2) 119897
times (119909 minus
radic3120588120572120573 (3 + 411989721205832)
120588 (3 + 411989721205832)
119905)
minus
3120573
16 (120573 + 1)
coth2radic120573 (3120573 + 6)
2 (120573 + 2) 119897
times (119909 minus
radic3120588120572120573 (3 + 411989721205832)
120588 (3 + 411989721205832)
119905)
1199086(119909 119905) = ∓
9120573
4radicminus2411989721205832+ 36120573 + 18
plusmn
9120573
8radicminus2411989721205832+ 36120573 + 18
tanh2radic120573 (3120573 + 6)
2 (120573 + 2) 119897
times (119909 minus
radic3120588120572120573 (3 + 411989721205832)
120588 (3 + 411989721205832)
119905)
plusmn
9120573
8radicminus2411989721205832+ 36120573 + 18
coth2radic120573 (3120573 + 6)
2 (120573 + 2) 119897
times (119909 minus
radic3120588120572120573 (3 + 411989721205832)
120588 (3 + 411989721205832)
119905)
(26)
33 The Third Coupled We now examine the coupled equa-tions
119906119905119905= 120573119906119909119909minus
120573
2
(1199062)119909119909+
1
2
(1199082)119909119909+ 1205722119906119909119909119905119905
119908119905119905= (119906119908)
119909119909+ 1205722119908119909119909119905119905
(27)
Using the wave variable 120585 = 119909 minus119881119905 the system (27) is carriedto a system of ODEs Integrating these equations twice andneglecting constants of integration we find
2 (1198812minus 120573)119880 + 120573119880
2minus 120573119882
2minus 21205722119881211988010158401015840= 0
1198812119882minus119880119882 minus 120572
2119881211988210158401015840= 0
(28)
Balancing 1205731198822 with 21205722119881211988010158401015840 and 119881211988210158401015840 with 119880119882 in (28)gives
2119873 = 119872 + 2
2 + 119873 = 119872 +119873
(29)
so that
119872 = 119873 = 2 (30)
As a result we seek solutions to (27) in the form
119880(120583120585) = 119878 (119884) =
2
sum
119896=0
119886119896119884119896+
2
sum
119896=1
119887119896119884minus119896
119882 (120583120585) = 119878 (119884) =
2
sum
119896=0
119888119896119884119896+
2
sum
119896=1
119889119896119884119896
(31)
Substitution of (31) into (28) leads to an algebraic system inthe unknowns 119886
0 1198861 1198862 1198871 1198872 1198880 1198881 1198882 1198891 1198892 119881 and 120583
8 Journal of Applied Mathematics
Collecting the coefficients of 119884 and solving the resultingsystem we find the following sets of solutions
1198860=
120573 (3120573 + 8)
8 (120573 + 1)
1198862=
minus31205732
16 (120573 + 1)
1198872= 1198862 119888
0= minus2119889
2 119888
2= 1198892
1198892=
plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573
16
119881 =
radic120573 (1612057221205832+ 1)
1612057221205832+ 1
120583 =
radic120573 (120573 + 2)
4120572 (120573 + 2)
1198861= 1198871= 1198881= 1198891= 0
1198860=
120573 (120573 + 8)
8 (120573 + 1)
1198862=
31205732
16 (120573 + 1)
1198872= 1198862
1198880=
21198892
3
1198882= 1198892
1198892=
plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573
16
119881 =
radicminus120573 (1612057221205832minus 1)
1612057221205832minus 1
120583 =
radicminus120573 (120573 + 2)
4120572 (120573 + 2)
1198861= 1198871= 1198881= 1198891= 0
1198860=
minus120573 (120573 minus 4)
4 (120573 + 1)
1198862=
31205732
4 (120573 + 1)
1198880= minus
1198882
3
1198882=
plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573
4
119881 =
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
120583 =
radicminus120573 (120573 + 2)
2120572 (120573 + 2)
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860=
120573 (3120573 + 4)
4 (120573 + 1)
1198862=
minus31205732
4 (120573 + 1)
1198880= minus2
1198882=
plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573
4
119881 =
radic120573 (412057221205832+ 1)
412057221205832+ 1
120583 =
radic120573 (120573 + 2)
2120572 (120573 + 2)
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860=
120573
120573 + 1
1198862=
31205732
2 (31205732+ 7120573 + 4)
1198872= 1198862 119888
2= minus1198892
1198892=
plusmn3radicminus421198814120573 minus 58119881
4minus 21198812120573 minus 120119881
2+ 120120573
8
119881 =
radic5120573 (812057221205832+ 3120573 + 5)
812057221205832+ 3120573 + 5
120583 =
radicminus120573 (5120573 + 4)
2120572 (5120573 + 4)
1198861= 1198871= 1198880= 1198881= 1198891= 0
1198860=
120573
120573 + 1
1198862=
minus120573
2 (120573 + 1)
1198872= 1198862
1198882= minus1198892 119889
2=
plusmnradic91198812+ 6
2
119881 =
radic3120573 (120573 + 1)
3 (120573 + 1)
120583 =
minus1
2120572
1198861= 1198871= 1198880= 1198881= 1198891= 0
(32)
Consequently we obtain the following solutions
1199061(119909 119905) =
120573 (3120573 + 8)
8 (120573 + 1)
minus
31205732
16 (120573 + 1)
tanh2radic120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radic120573 (1612057221205832+ 1)
1612057221205832+ 1
119905)
minus
31205732
16 (120573 + 1)
coth2radic120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radic120573 (1612057221205832+ 1)
1612057221205832+ 1
119905)
1199081(119909 119905) =
∓3radicminus41198814minus 41198812+ 21205732+ 4120573
8
plusmn
3radicminus41198814minus 41198812+ 21205732+ 4120573
16
tanh2radic120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radic120573 (1612057221205832+ 1)
1612057221205832+ 1
119905)
plusmn
3radicminus41198814minus 41198812+ 21205732+ 4120573
16
coth2radic120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radic120573 (1612057221205832+ 1)
1612057221205832+ 1
119905)
Journal of Applied Mathematics 9
1199062(119909 119905) =
120573 (120573 + 8)
8 (120573 + 1)
+
31205732
16 (120573 + 1)
tanh2radicminus120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radicminus120573 (1612057221205832minus 1)
1612057221205832minus 1
119905)
+
31205732
16 (120573 + 1)
coth2radicminus120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radicminus120573 (1612057221205832minus 1)
1612057221205832minus 1
119905)
1199082(119909 119905) = plusmn
radicminus41198814minus 41198812+ 21205732+ 4120573
8
plusmn
3radicminus41198814minus 41198812+ 21205732+ 4120573
16
tanh2radicminus120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radicminus120573 (1612057221205832minus 1)
1612057221205832minus 1
119905)
plusmn
3radicminus41198814minus 41198812+ 21205732+ 4120573
16
coth2radicminus120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radicminus120573 (1612057221205832minus 1)
1612057221205832minus 1
119905)
1199063(119909 119905) =
minus120573 (120573 minus 4)
4 (120573 + 1)
+
31205732
4 (120573 + 1)
tanh2radicminus120573 (120573 + 2)
2120572 (120573 + 2)
times (119909 minus
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
119905)
1199083(119909 119905) = ∓
radicminus41198814minus 41198812+ 21205732+ 4120573
4
plusmn
3radicminus41198814minus 41198812+ 21205732+ 4120573
4
tanhradicminus120573 (120573 + 2)
2120572 (120573 + 2)
times (119909 minus
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
119905)
1199064(119909 119905) =
120573 (3120573 + 4)
4 (120573 + 1)
minus
31205732
4 (120573 + 1)
tanh2radic120573 (120573 + 2)
2120572 (120573 + 2)
times (119909 minus
radic120573 (412057221205832+ 1)
412057221205832+ 1
119905)
1199084(119909 119905) = ∓
3radicminus41198814minus 41198812+ 21205732+ 4120573
4
plusmn
3radicminus41198814minus 41198812+ 21205732+ 4120573
4
tanh2radic120573 (120573 + 2)
2120572 (120573 + 2)
times (119909 minus
radic120573 (412057221205832+ 1)
412057221205832+ 1
119905)
1199065(119909 119905) =
120573
120573 + 1
+
31205732
2 (31205732+ 7120573 + 4)
tanh2radicminus120573 (5120573 + 4)
2120572 (5120573 + 4)
times (119909 minus
radic5120573 (812057221205832+ 3120573 + 5)
812057221205832+ 3120573 + 5
119905)
+
31205732
2 (31205732+ 7120573 + 4)
coth2radicminus120573 (5120573 + 4)
2120572 (5120573 + 4)
times (119909 minus
radic5120573 (812057221205832+ 3120573 + 5)
812057221205832+ 3120573 + 5
119905)
1199085(119909 119905)
= ∓
3radicminus421198814120573 minus 58119881
4minus 21198812120573 minus 120119881
2+ 120120573
8
times tanh2radicminus120573 (5120573 + 4)
2120572 (5120573 + 4)
times (119909 minus
radic5120573 (812057221205832+ 3120573 + 5)
812057221205832+ 3120573 + 5
119905)
plusmn
3radicminus421198814120573 minus 58119881
4minus 21198812120573 minus 120119881
2+ 120120573
8
10 Journal of Applied Mathematics
times coth2radicminus120573 (5120573 + 4)
2120572 (5120573 + 4)
times (119909 minus
radic5120573 (812057221205832+ 3120573 + 5)
812057221205832+ 3120573 + 5
119905)
1199066(119909 119905) =
120573
120573 + 1
minus
120573
2 (120573 + 1)
tanh2minus12120572
(119909 minus
radic3120573 (120573 + 1)
3 (120573 + 1)
119905)
minus
120573
2 (120573 + 1)
coth2minus12120572
(119909 minus
radic3120573 (120573 + 1)
3 (120573 + 1)
119905)
1199086(119909 119905)
= ∓
radic91198812+ 6
2
tanh2minus12120572
(119909 minus
radic3120573 (120573 + 1)
3 (120573 + 1)
119905)
plusmn
radic91198812+ 6
2
coth2minus12120572
(119909 minus
radic3120573 (120573 + 1)
3 (120573 + 1)
119905)
(33)
34 The Fourth Coupled We now consider the coupled IBqequations
119906119905119905= 119906119909119909minus 120573(119906
2)119909119909+ (1199082)119909119909+ 119906119909119909119905119905
119908119905119905= 119908119909119909+ 2(119906119908)
119909119909+ 119908119909119909119905119905
(34)
Using the wave variable 120585 = 119909 minus 119881119905 (34) can be converted toa system of ODEs
(1198812minus 1)119880 + 120573119880
2minus1198822minus 119881211988010158401015840= 0
(1198812minus 1)119882 minus 2119880119882 minus 119881
211988210158401015840= 0
(35)
Balancing1198822 with 11988010158401015840 and 119880119882 with11988210158401015840 in (35) gives
2119873 = 119872 + 2
119872 + 119873 = 119873 + 2
(36)
so that
119872 = 119873 = 2 (37)
The tanh-coth method substitutes the finite expansions
119880(120583120585) = 119878 (119884) =
2
sum
119896=0
119886119896119884119896+
2
sum
119896=1
119887119896119884minus119896
119882 (120583120585) = 119878 (119884) =
2
sum
119896=0
119888119896119884119896+
2
sum
119896=1
119889119896119884119896
(38)
In (35) collecting the coefficients of each power of 119884 andsolving the resulting system we obtain
1198860=
minus21205832
161205832+ 1
1198862=
minus31205832
161205832+ 1
1198862= 1198872
1198880=
21198892
3
1198882= 1198892
1198892=
plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
119881 =
1
radic161205832+ 1
1198861= 1198871= 1198881= 1198891= 0
1198860=
minus61205832
161205832minus 1
1198862=
31205832
161205832minus 1
1198862= 1198872
1198880= minus2119889
2 119888
2= 1198892
1198892=
plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
119881 =
1
radicminus161205832+ 1
1198861= 1198871= 1198881= 1198891= 0
1198860=
1205832
41205832+ 1
1198862=
minus31205832
41205832+ 1
1198880=
minus1198882
3
1198882=
plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
119881 =
1
radic41205832+ 1
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860=
minus31205832
41205832minus 1
1198862=
31205832
41205832minus 1
1198880= minus1198882
1198882=
plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
119881 =
1
radicminus41205832+ 1
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860=
minus31205832
41205832minus 1
1198872=
31205832
41205832minus 1
1198880= minus1198892
1198892=
plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
119881 =
1
radicminus41205832+ 1
1198861= 1198862= 1198871= 1198881= 1198882= 1198891= 0
(39)
Journal of Applied Mathematics 11
where 120583 is left as a free parameter Consequently we obtainthe following travelling wave solutions
1199061(119909 119905) = minus
21205832
161205832+ 1
minus
31205832
161205832+ 1
tanh2120583(119909 minus 1
radic161205832+ 1
119905)
minus
31205832
161205832+ 1
coth2120583(119909 minus 1
radic161205832+ 1
119905)
1199081(119909 119905) = plusmn
(119881 + 1) (119881 minus 1)radic120573 + 2
8
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
tanh2120583
times(119909 minus
1
radic161205832+ 1
119905)
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
coth2120583
times(119909 minus
1
radic161205832+ 1
119905)
1199062(119909 119905) = minus
61205832
161205832minus 1
+
31205832
161205832minus 1
tanh2120583(119909 minus 1
radicminus161205832+ 1
119905)
+
31205832
161205832minus 1
coth2120583(119909 minus 1
radicminus161205832+ 1
119905)
1199082(119909 119905) = ∓
3 (119881 + 1) (119881 minus 1)radic120573 + 2
8
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
tanh2120583
times(119909 minus
1
radicminus161205832+ 1
119905)
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
coth2120583
times(119909 minus
1
radicminus161205832+ 1
119905)
1199063(119909 119905) =
1205832
41205832+ 1
minus
31205832
41205832+ 1
tanh2120583(119909 minus 1
radic41205832+ 1
119905)
1199083(119909 119905) = ∓
(119881 + 1) (119881 minus 1)radic120573 + 2
4
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
times tanh2120583(119909 minus 1
radic41205832+ 1
119905)
1199064(119909 119905) = minus
31205832
41205832minus 1
+
31205832
41205832minus 1
tanh2120583(119909 minus 1
radicminus41205832+ 1
119905)
1199084(119909 119905) = ∓
3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
times tanh2120583(119909 minus 1
radicminus41205832+ 1
119905)
1199065(119909 119905) = minus
31205832
41205832minus 1
+
31205832
41205832minus 1
coth2120583(119909 minus 1
radicminus41205832+ 1
119905)
1199085(119909 119905) = ∓
3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
plusmn 3
(119881 + 1) (119881 minus 1)radic120573 + 2
4
times coth2120583(119909 minus 1
radicminus41205832+ 1
119905)
(40)
35 The Fifth Coupled We finally consider the coupled IBqequations
119906119905119905= 120573119906119909119909minus
120573
2
(1199062)119909119909+
120573
2
(1199082)119909119909+ 1205722119906119909119909119905119905
119908119905119905= 120574(119906119908)
119909119909+ 120588(119908
2)119909119909+ 1205722119908119909119909119905119905
(41)
12 Journal of Applied Mathematics
This system of equations (41) can be converted to the follow-ing system of ODEs by using the wave variable 120585 = 119909 minus 119881119905
(21198812minus 2120573)119880 + 120573119880
2minus 120573119882
2minus 21205722119881211988010158401015840= 0
1198812119882minus 120574119880119882 minus 120588119882
2minus 1205722119881211988210158401015840= 0
(42)
Using the balancing procedure in (42) gives
2119872 = 119872 +119873
2119873 = 119872 +119873
(43)
so that
119872 = 119873 = 2 (44)
We consider solutions in the following form
119880 (120583120585) = 119878 (119884) =
2
sum
119896=0
119886119896119884119896+
2
sum
119896=1
119887119896119884minus119896
119882 (120583120585) = 119878 (119884) =
2
sum
119896=0
119888119896119884119896+
2
sum
119896=1
119889119896119884119896
(45)
Substituting (45) into (42) collecting the coefficients of 119884and solving the resulting system we find the following setsof solutions
1198860= (minus120573 (8120588119888
21205732120574 minus 8120573120574
21205881198882minus 18120588
21205732+ 24120573120588
31198882
+1212057321205742minus 16120574
31205881198882+ 24120588
31198882120574 minus 3120574120573
3))
times (4 (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882+ 312058821205733
+ 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882minus 312057331205742minus 312057431205732))
minus1
1198862=
minus (41205881198882120573 + 4119888
2120588120574 minus 3120573
2)
4120574 (120583 + 120573)
1198880= (120573(120588
212057321198882+ 120574211988821205732+ 12057311988821205743+ 2120574212058821198882
+312057412058821205731198882minus
3
2
1205881205732120574 minus
3
4
1205881205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882+ 312058821205733
+ 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882minus 312057331205742minus 312057431205732)
minus1
1198882= (plusmn3 (minus 120573120588 + (minus120588
21205732+ 41205743120573 minus 4120574
21198814
minus 412057431198812+ 212057321205742
minus41205882120574120573 + 4120588
21198814+ 412058821198812120574)
12
))
times (4 (1205742minus 1205882))
minus1
119881 =
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
120583 =
radicminus120573 (2120574 + 120573)
2120572 (2120574 + 120573)
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860= (120573(6120588119888
21205732120574 + 10120573120574
21205881198882+
3
2
12058821205732+ 2120573120588
31198882
+312057321205742+ 412057431205881198882+ 212058831198882120574 +
9
4
1205741205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882minus 312058821205733
minus 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882+ 312057331205742+ 312057431205732)
minus1
1198862=
minus (41205881198882120573 + 4119888
2120588120574 + 3120573
2)
4120574 (120583 + 120573)
1198880= (120573(minus 3120588
212057321198882minus 3120574211988821205732minus 312057311988821205743
minus6120574212058821198882minus 9120574120588
21205731198882minus
9
2
1205881205732120574 minus
9
4
1205881205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882
minus 312058821205733minus 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882+ 312057331205742+ 312057431205732)
minus1
1198882= (plusmn3 (120573120588 + (minus120588
21205732+ 41205743120573 minus 4120574
21198814
minus 412057431198812+ 212057321205742minus 41205882120574120573
+412058821198814+ 412058821198812120574)
12
))
times (4 (1205742minus 1205882))
minus1
119881 =
radic120573 (412057221205832+ 1)
412057221205832+ 1
120583 =
radic120573 (2120574 + 120573)
2120572 (2120574 + 120573)
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
(46)
Journal of Applied Mathematics 13
In view of this we obtain the following travelling wavesolutions
1199061(119909 119905) = (minus120573 (8120588119888
21205732120574 minus 8120573120574
21205881198882minus 18120588
21205732+ 24120573120588
31198882
+1212057321205742minus 16120574
31205881198882+ 24120588
31198882120574 minus 3120574120573
3))
times (4 (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882+ 312058821205733
+ 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882minus 312057331205742minus 312057431205732))
minus1
minus
(41205881198882120573 + 4119888
2120588120574 minus 3120573
2)
4120574 (120583 + 120573)
tanh2radicminus120573 (2120574 + 120573)
2120572 (2120574 + 120573)
times(119909 minus
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
119905)
1199081(119909 119905) = (120573 (120588
212057321198882+ 120574211988821205732+ 12057311988821205743+ 2120574212058821198882
+312057412058821205731198882minus
3
2
1205881205732120574 minus
3
4
1205881205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882+ 312058821205733
+ 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882minus 312057331205742minus 312057431205732)
minus1
plusmn (3 (minus 120573120588 + (minus12058821205732+ 41205743120573 minus 4120574
21198814
minus 412057431198812+ 212057321205742minus 41205882120574120573
+412058821198814+ 412058821198812120574)
12
))
times (4 (1205742minus 1205882))
minus1
times tanh2radicminus120573 (2120574 + 120573)
2120572 (2120574 + 120573)
times(119909 minus
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
119905)
1199062(119909 119905) = (120573 (6120588119888
21205732120574 + 10120573120574
21205881198882+
3
2
12058821205732+ 2120573120588
31198882
+312057321205742+ 412057431205881198882+ 212058831198882120574 +
9
4
1205741205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882minus 312058821205733
minus 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882+ 312057331205742+ 312057431205732)
minus1
minus
(41205881198882120573 + 4119888
2120588120574 + 3120573
2)
4120574 (120583 + 120573)
tanh2radic120573 (2120574 + 120573)
2120572 (2120574 + 120573)
times(119909 minus
radic120573 (412057221205832+ 1)
412057221205832+ 1
119905)
1199082(119909 119905) = (120573 (minus 3120588
212057321198882minus 3120574211988821205732minus 312057311988821205743
minus6120574212058821198882minus 9120574120588
21205731198882minus
9
2
1205881205732120574 minus
9
4
1205881205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882minus 312058821205733
minus 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882+ 312057331205742+ 312057431205732)
minus1
plusmn (3 (120573120588 + (minus12058821205732+ 41205743120573 minus 4120574
21198814
minus 412057431198812+ 212057321205742minus 41205882120574120573
+412058821198814+ 412058821198812120574)
12
))
times (4 (1205742minus 1205882))
minus1
times tanh2radic120573 (2120574 + 120573)
2120572 (2120574 + 120573)
(119909 minus
radic120573 (412057221205832+ 1)
412057221205832+ 1
119905)
(47)
4 Conclusion
In summary the tanh-coth method has been successfullyimplemented to find new travelling wave solutions for severalkinds of two coupled improved Boussinesq equations Thesolutions that have been found can be seen as an expansionand verification of the previously known data However thisstudy confirmed again the belief that the tanh-coth methodis a powerful and reliable technique to handle nonlineardispersive-dissipative equations and system of equationsThroughout the work Maple has been used to overcome thetedious algebraic calculations
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] A De Godefroy ldquoBlow up of solutions of a generalized Boussi-nesq equationrdquo IMA Journal of AppliedMathematics vol 60 no2 pp 123ndash138 1998
14 Journal of Applied Mathematics
[2] S Wang and M Li ldquoThe Cauchy problem for coupled IMBqequationsrdquo IMA Journal of Applied Mathematics vol 74 no 5pp 726ndash740 2009
[3] N Duruk H A Erbay and A Erkip ldquoBlow-up and globalexistence for a general class of nonlocal nonlinear coupled waveequationsrdquo Journal of Differential Equations vol 250 no 3 pp1448ndash1459 2011
[4] G Chen and H Zhang ldquoInitial boundary value problem for asystem of generalized IMBq equationsrdquoMathematical Methodsin the Applied Sciences vol 27 no 5 pp 497ndash518 2004
[5] P Rosenau ldquoDynamics of dense latticesrdquo Physical Review B vol36 no 11 pp 5868ndash5876 1987
[6] P L Christiansen P S Lomdahl andVMuto ldquoOn aToda latticemodel with a transversal degree of freedomrdquo Nonlinearity vol4 no 2 pp 477ndash501 1991
[7] S K Turitsyn ldquoOn a Toda lattice model with a transversaldegree of freedom Sufficient criterion of blow-up in thecontinuum limitrdquo Physics Letters A vol 173 no 3 pp 267ndash2691993
[8] R L Pego P Smereka and M I Weinstein ldquoOscillatoryinstability of solitary waves in a continuum model of latticevibrationsrdquo Nonlinearity vol 8 no 6 pp 921ndash941 1995
[9] X Fan and L Tian ldquoCompacton solutions and multiple com-pacton solutions for a continuum Toda lattice modelrdquo ChaosSolitons and Fractals vol 29 no 4 pp 882ndash894 2006
[10] A M Wazwaz ldquoThe Hirotarsquos bilinear method and the tanh-coth method for multiple-soliton solutions of the Sawada-Kotera-Kadomtsev-Petviashvili equationrdquoAppliedMathematicsand Computation vol 200 no 1 pp 160ndash166 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 7
times(119909 minus
radicminus3120588120572120573 (minus3 + 411989721205832)
120588 (minus3 + 411989721205832)
119905)
+
3120573
16 (120573 + 1)
coth2radicminus120573 (3120573 + 6)
2 (120573 + 2) 119897
times (119909 minus
radicminus3120588120572120573 (minus3 + 411989721205832)
120588 (minus3 + 411989721205832)
119905)
1199085(119909 119905) = plusmn
3120573
4radic2411989721205832+ 36120573 + 18
plusmn
9120573
8radic2411989721205832+ 36120573 + 18
tanh2radicminus120573 (3120573 + 6)
2 (120573 + 2) 119897
times (119909 minus
radicminus3120588120572120573 (minus3 + 411989721205832)
120588 (minus3 + 411989721205832)
119905)
plusmn
9120573
8radic2411989721205832+ 36120573 + 18
coth2radicminus120573 (3120573 + 6)
2 (120573 + 2) 119897
times (119909 minus
radicminus3120588120572120573 (minus3 + 411989721205832)
120588 (minus3 + 411989721205832)
119905)
1199066(119909 119905) =
3 (120573 + 8)
8 (120573 + 1)
minus
3120573
16 (120573 + 1)
tanh2radic120573 (3120573 + 6)
2 (120573 + 2) 119897
times (119909 minus
radic3120588120572120573 (3 + 411989721205832)
120588 (3 + 411989721205832)
119905)
minus
3120573
16 (120573 + 1)
coth2radic120573 (3120573 + 6)
2 (120573 + 2) 119897
times (119909 minus
radic3120588120572120573 (3 + 411989721205832)
120588 (3 + 411989721205832)
119905)
1199086(119909 119905) = ∓
9120573
4radicminus2411989721205832+ 36120573 + 18
plusmn
9120573
8radicminus2411989721205832+ 36120573 + 18
tanh2radic120573 (3120573 + 6)
2 (120573 + 2) 119897
times (119909 minus
radic3120588120572120573 (3 + 411989721205832)
120588 (3 + 411989721205832)
119905)
plusmn
9120573
8radicminus2411989721205832+ 36120573 + 18
coth2radic120573 (3120573 + 6)
2 (120573 + 2) 119897
times (119909 minus
radic3120588120572120573 (3 + 411989721205832)
120588 (3 + 411989721205832)
119905)
(26)
33 The Third Coupled We now examine the coupled equa-tions
119906119905119905= 120573119906119909119909minus
120573
2
(1199062)119909119909+
1
2
(1199082)119909119909+ 1205722119906119909119909119905119905
119908119905119905= (119906119908)
119909119909+ 1205722119908119909119909119905119905
(27)
Using the wave variable 120585 = 119909 minus119881119905 the system (27) is carriedto a system of ODEs Integrating these equations twice andneglecting constants of integration we find
2 (1198812minus 120573)119880 + 120573119880
2minus 120573119882
2minus 21205722119881211988010158401015840= 0
1198812119882minus119880119882 minus 120572
2119881211988210158401015840= 0
(28)
Balancing 1205731198822 with 21205722119881211988010158401015840 and 119881211988210158401015840 with 119880119882 in (28)gives
2119873 = 119872 + 2
2 + 119873 = 119872 +119873
(29)
so that
119872 = 119873 = 2 (30)
As a result we seek solutions to (27) in the form
119880(120583120585) = 119878 (119884) =
2
sum
119896=0
119886119896119884119896+
2
sum
119896=1
119887119896119884minus119896
119882 (120583120585) = 119878 (119884) =
2
sum
119896=0
119888119896119884119896+
2
sum
119896=1
119889119896119884119896
(31)
Substitution of (31) into (28) leads to an algebraic system inthe unknowns 119886
0 1198861 1198862 1198871 1198872 1198880 1198881 1198882 1198891 1198892 119881 and 120583
8 Journal of Applied Mathematics
Collecting the coefficients of 119884 and solving the resultingsystem we find the following sets of solutions
1198860=
120573 (3120573 + 8)
8 (120573 + 1)
1198862=
minus31205732
16 (120573 + 1)
1198872= 1198862 119888
0= minus2119889
2 119888
2= 1198892
1198892=
plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573
16
119881 =
radic120573 (1612057221205832+ 1)
1612057221205832+ 1
120583 =
radic120573 (120573 + 2)
4120572 (120573 + 2)
1198861= 1198871= 1198881= 1198891= 0
1198860=
120573 (120573 + 8)
8 (120573 + 1)
1198862=
31205732
16 (120573 + 1)
1198872= 1198862
1198880=
21198892
3
1198882= 1198892
1198892=
plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573
16
119881 =
radicminus120573 (1612057221205832minus 1)
1612057221205832minus 1
120583 =
radicminus120573 (120573 + 2)
4120572 (120573 + 2)
1198861= 1198871= 1198881= 1198891= 0
1198860=
minus120573 (120573 minus 4)
4 (120573 + 1)
1198862=
31205732
4 (120573 + 1)
1198880= minus
1198882
3
1198882=
plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573
4
119881 =
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
120583 =
radicminus120573 (120573 + 2)
2120572 (120573 + 2)
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860=
120573 (3120573 + 4)
4 (120573 + 1)
1198862=
minus31205732
4 (120573 + 1)
1198880= minus2
1198882=
plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573
4
119881 =
radic120573 (412057221205832+ 1)
412057221205832+ 1
120583 =
radic120573 (120573 + 2)
2120572 (120573 + 2)
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860=
120573
120573 + 1
1198862=
31205732
2 (31205732+ 7120573 + 4)
1198872= 1198862 119888
2= minus1198892
1198892=
plusmn3radicminus421198814120573 minus 58119881
4minus 21198812120573 minus 120119881
2+ 120120573
8
119881 =
radic5120573 (812057221205832+ 3120573 + 5)
812057221205832+ 3120573 + 5
120583 =
radicminus120573 (5120573 + 4)
2120572 (5120573 + 4)
1198861= 1198871= 1198880= 1198881= 1198891= 0
1198860=
120573
120573 + 1
1198862=
minus120573
2 (120573 + 1)
1198872= 1198862
1198882= minus1198892 119889
2=
plusmnradic91198812+ 6
2
119881 =
radic3120573 (120573 + 1)
3 (120573 + 1)
120583 =
minus1
2120572
1198861= 1198871= 1198880= 1198881= 1198891= 0
(32)
Consequently we obtain the following solutions
1199061(119909 119905) =
120573 (3120573 + 8)
8 (120573 + 1)
minus
31205732
16 (120573 + 1)
tanh2radic120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radic120573 (1612057221205832+ 1)
1612057221205832+ 1
119905)
minus
31205732
16 (120573 + 1)
coth2radic120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radic120573 (1612057221205832+ 1)
1612057221205832+ 1
119905)
1199081(119909 119905) =
∓3radicminus41198814minus 41198812+ 21205732+ 4120573
8
plusmn
3radicminus41198814minus 41198812+ 21205732+ 4120573
16
tanh2radic120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radic120573 (1612057221205832+ 1)
1612057221205832+ 1
119905)
plusmn
3radicminus41198814minus 41198812+ 21205732+ 4120573
16
coth2radic120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radic120573 (1612057221205832+ 1)
1612057221205832+ 1
119905)
Journal of Applied Mathematics 9
1199062(119909 119905) =
120573 (120573 + 8)
8 (120573 + 1)
+
31205732
16 (120573 + 1)
tanh2radicminus120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radicminus120573 (1612057221205832minus 1)
1612057221205832minus 1
119905)
+
31205732
16 (120573 + 1)
coth2radicminus120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radicminus120573 (1612057221205832minus 1)
1612057221205832minus 1
119905)
1199082(119909 119905) = plusmn
radicminus41198814minus 41198812+ 21205732+ 4120573
8
plusmn
3radicminus41198814minus 41198812+ 21205732+ 4120573
16
tanh2radicminus120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radicminus120573 (1612057221205832minus 1)
1612057221205832minus 1
119905)
plusmn
3radicminus41198814minus 41198812+ 21205732+ 4120573
16
coth2radicminus120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radicminus120573 (1612057221205832minus 1)
1612057221205832minus 1
119905)
1199063(119909 119905) =
minus120573 (120573 minus 4)
4 (120573 + 1)
+
31205732
4 (120573 + 1)
tanh2radicminus120573 (120573 + 2)
2120572 (120573 + 2)
times (119909 minus
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
119905)
1199083(119909 119905) = ∓
radicminus41198814minus 41198812+ 21205732+ 4120573
4
plusmn
3radicminus41198814minus 41198812+ 21205732+ 4120573
4
tanhradicminus120573 (120573 + 2)
2120572 (120573 + 2)
times (119909 minus
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
119905)
1199064(119909 119905) =
120573 (3120573 + 4)
4 (120573 + 1)
minus
31205732
4 (120573 + 1)
tanh2radic120573 (120573 + 2)
2120572 (120573 + 2)
times (119909 minus
radic120573 (412057221205832+ 1)
412057221205832+ 1
119905)
1199084(119909 119905) = ∓
3radicminus41198814minus 41198812+ 21205732+ 4120573
4
plusmn
3radicminus41198814minus 41198812+ 21205732+ 4120573
4
tanh2radic120573 (120573 + 2)
2120572 (120573 + 2)
times (119909 minus
radic120573 (412057221205832+ 1)
412057221205832+ 1
119905)
1199065(119909 119905) =
120573
120573 + 1
+
31205732
2 (31205732+ 7120573 + 4)
tanh2radicminus120573 (5120573 + 4)
2120572 (5120573 + 4)
times (119909 minus
radic5120573 (812057221205832+ 3120573 + 5)
812057221205832+ 3120573 + 5
119905)
+
31205732
2 (31205732+ 7120573 + 4)
coth2radicminus120573 (5120573 + 4)
2120572 (5120573 + 4)
times (119909 minus
radic5120573 (812057221205832+ 3120573 + 5)
812057221205832+ 3120573 + 5
119905)
1199085(119909 119905)
= ∓
3radicminus421198814120573 minus 58119881
4minus 21198812120573 minus 120119881
2+ 120120573
8
times tanh2radicminus120573 (5120573 + 4)
2120572 (5120573 + 4)
times (119909 minus
radic5120573 (812057221205832+ 3120573 + 5)
812057221205832+ 3120573 + 5
119905)
plusmn
3radicminus421198814120573 minus 58119881
4minus 21198812120573 minus 120119881
2+ 120120573
8
10 Journal of Applied Mathematics
times coth2radicminus120573 (5120573 + 4)
2120572 (5120573 + 4)
times (119909 minus
radic5120573 (812057221205832+ 3120573 + 5)
812057221205832+ 3120573 + 5
119905)
1199066(119909 119905) =
120573
120573 + 1
minus
120573
2 (120573 + 1)
tanh2minus12120572
(119909 minus
radic3120573 (120573 + 1)
3 (120573 + 1)
119905)
minus
120573
2 (120573 + 1)
coth2minus12120572
(119909 minus
radic3120573 (120573 + 1)
3 (120573 + 1)
119905)
1199086(119909 119905)
= ∓
radic91198812+ 6
2
tanh2minus12120572
(119909 minus
radic3120573 (120573 + 1)
3 (120573 + 1)
119905)
plusmn
radic91198812+ 6
2
coth2minus12120572
(119909 minus
radic3120573 (120573 + 1)
3 (120573 + 1)
119905)
(33)
34 The Fourth Coupled We now consider the coupled IBqequations
119906119905119905= 119906119909119909minus 120573(119906
2)119909119909+ (1199082)119909119909+ 119906119909119909119905119905
119908119905119905= 119908119909119909+ 2(119906119908)
119909119909+ 119908119909119909119905119905
(34)
Using the wave variable 120585 = 119909 minus 119881119905 (34) can be converted toa system of ODEs
(1198812minus 1)119880 + 120573119880
2minus1198822minus 119881211988010158401015840= 0
(1198812minus 1)119882 minus 2119880119882 minus 119881
211988210158401015840= 0
(35)
Balancing1198822 with 11988010158401015840 and 119880119882 with11988210158401015840 in (35) gives
2119873 = 119872 + 2
119872 + 119873 = 119873 + 2
(36)
so that
119872 = 119873 = 2 (37)
The tanh-coth method substitutes the finite expansions
119880(120583120585) = 119878 (119884) =
2
sum
119896=0
119886119896119884119896+
2
sum
119896=1
119887119896119884minus119896
119882 (120583120585) = 119878 (119884) =
2
sum
119896=0
119888119896119884119896+
2
sum
119896=1
119889119896119884119896
(38)
In (35) collecting the coefficients of each power of 119884 andsolving the resulting system we obtain
1198860=
minus21205832
161205832+ 1
1198862=
minus31205832
161205832+ 1
1198862= 1198872
1198880=
21198892
3
1198882= 1198892
1198892=
plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
119881 =
1
radic161205832+ 1
1198861= 1198871= 1198881= 1198891= 0
1198860=
minus61205832
161205832minus 1
1198862=
31205832
161205832minus 1
1198862= 1198872
1198880= minus2119889
2 119888
2= 1198892
1198892=
plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
119881 =
1
radicminus161205832+ 1
1198861= 1198871= 1198881= 1198891= 0
1198860=
1205832
41205832+ 1
1198862=
minus31205832
41205832+ 1
1198880=
minus1198882
3
1198882=
plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
119881 =
1
radic41205832+ 1
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860=
minus31205832
41205832minus 1
1198862=
31205832
41205832minus 1
1198880= minus1198882
1198882=
plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
119881 =
1
radicminus41205832+ 1
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860=
minus31205832
41205832minus 1
1198872=
31205832
41205832minus 1
1198880= minus1198892
1198892=
plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
119881 =
1
radicminus41205832+ 1
1198861= 1198862= 1198871= 1198881= 1198882= 1198891= 0
(39)
Journal of Applied Mathematics 11
where 120583 is left as a free parameter Consequently we obtainthe following travelling wave solutions
1199061(119909 119905) = minus
21205832
161205832+ 1
minus
31205832
161205832+ 1
tanh2120583(119909 minus 1
radic161205832+ 1
119905)
minus
31205832
161205832+ 1
coth2120583(119909 minus 1
radic161205832+ 1
119905)
1199081(119909 119905) = plusmn
(119881 + 1) (119881 minus 1)radic120573 + 2
8
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
tanh2120583
times(119909 minus
1
radic161205832+ 1
119905)
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
coth2120583
times(119909 minus
1
radic161205832+ 1
119905)
1199062(119909 119905) = minus
61205832
161205832minus 1
+
31205832
161205832minus 1
tanh2120583(119909 minus 1
radicminus161205832+ 1
119905)
+
31205832
161205832minus 1
coth2120583(119909 minus 1
radicminus161205832+ 1
119905)
1199082(119909 119905) = ∓
3 (119881 + 1) (119881 minus 1)radic120573 + 2
8
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
tanh2120583
times(119909 minus
1
radicminus161205832+ 1
119905)
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
coth2120583
times(119909 minus
1
radicminus161205832+ 1
119905)
1199063(119909 119905) =
1205832
41205832+ 1
minus
31205832
41205832+ 1
tanh2120583(119909 minus 1
radic41205832+ 1
119905)
1199083(119909 119905) = ∓
(119881 + 1) (119881 minus 1)radic120573 + 2
4
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
times tanh2120583(119909 minus 1
radic41205832+ 1
119905)
1199064(119909 119905) = minus
31205832
41205832minus 1
+
31205832
41205832minus 1
tanh2120583(119909 minus 1
radicminus41205832+ 1
119905)
1199084(119909 119905) = ∓
3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
times tanh2120583(119909 minus 1
radicminus41205832+ 1
119905)
1199065(119909 119905) = minus
31205832
41205832minus 1
+
31205832
41205832minus 1
coth2120583(119909 minus 1
radicminus41205832+ 1
119905)
1199085(119909 119905) = ∓
3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
plusmn 3
(119881 + 1) (119881 minus 1)radic120573 + 2
4
times coth2120583(119909 minus 1
radicminus41205832+ 1
119905)
(40)
35 The Fifth Coupled We finally consider the coupled IBqequations
119906119905119905= 120573119906119909119909minus
120573
2
(1199062)119909119909+
120573
2
(1199082)119909119909+ 1205722119906119909119909119905119905
119908119905119905= 120574(119906119908)
119909119909+ 120588(119908
2)119909119909+ 1205722119908119909119909119905119905
(41)
12 Journal of Applied Mathematics
This system of equations (41) can be converted to the follow-ing system of ODEs by using the wave variable 120585 = 119909 minus 119881119905
(21198812minus 2120573)119880 + 120573119880
2minus 120573119882
2minus 21205722119881211988010158401015840= 0
1198812119882minus 120574119880119882 minus 120588119882
2minus 1205722119881211988210158401015840= 0
(42)
Using the balancing procedure in (42) gives
2119872 = 119872 +119873
2119873 = 119872 +119873
(43)
so that
119872 = 119873 = 2 (44)
We consider solutions in the following form
119880 (120583120585) = 119878 (119884) =
2
sum
119896=0
119886119896119884119896+
2
sum
119896=1
119887119896119884minus119896
119882 (120583120585) = 119878 (119884) =
2
sum
119896=0
119888119896119884119896+
2
sum
119896=1
119889119896119884119896
(45)
Substituting (45) into (42) collecting the coefficients of 119884and solving the resulting system we find the following setsof solutions
1198860= (minus120573 (8120588119888
21205732120574 minus 8120573120574
21205881198882minus 18120588
21205732+ 24120573120588
31198882
+1212057321205742minus 16120574
31205881198882+ 24120588
31198882120574 minus 3120574120573
3))
times (4 (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882+ 312058821205733
+ 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882minus 312057331205742minus 312057431205732))
minus1
1198862=
minus (41205881198882120573 + 4119888
2120588120574 minus 3120573
2)
4120574 (120583 + 120573)
1198880= (120573(120588
212057321198882+ 120574211988821205732+ 12057311988821205743+ 2120574212058821198882
+312057412058821205731198882minus
3
2
1205881205732120574 minus
3
4
1205881205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882+ 312058821205733
+ 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882minus 312057331205742minus 312057431205732)
minus1
1198882= (plusmn3 (minus 120573120588 + (minus120588
21205732+ 41205743120573 minus 4120574
21198814
minus 412057431198812+ 212057321205742
minus41205882120574120573 + 4120588
21198814+ 412058821198812120574)
12
))
times (4 (1205742minus 1205882))
minus1
119881 =
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
120583 =
radicminus120573 (2120574 + 120573)
2120572 (2120574 + 120573)
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860= (120573(6120588119888
21205732120574 + 10120573120574
21205881198882+
3
2
12058821205732+ 2120573120588
31198882
+312057321205742+ 412057431205881198882+ 212058831198882120574 +
9
4
1205741205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882minus 312058821205733
minus 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882+ 312057331205742+ 312057431205732)
minus1
1198862=
minus (41205881198882120573 + 4119888
2120588120574 + 3120573
2)
4120574 (120583 + 120573)
1198880= (120573(minus 3120588
212057321198882minus 3120574211988821205732minus 312057311988821205743
minus6120574212058821198882minus 9120574120588
21205731198882minus
9
2
1205881205732120574 minus
9
4
1205881205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882
minus 312058821205733minus 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882+ 312057331205742+ 312057431205732)
minus1
1198882= (plusmn3 (120573120588 + (minus120588
21205732+ 41205743120573 minus 4120574
21198814
minus 412057431198812+ 212057321205742minus 41205882120574120573
+412058821198814+ 412058821198812120574)
12
))
times (4 (1205742minus 1205882))
minus1
119881 =
radic120573 (412057221205832+ 1)
412057221205832+ 1
120583 =
radic120573 (2120574 + 120573)
2120572 (2120574 + 120573)
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
(46)
Journal of Applied Mathematics 13
In view of this we obtain the following travelling wavesolutions
1199061(119909 119905) = (minus120573 (8120588119888
21205732120574 minus 8120573120574
21205881198882minus 18120588
21205732+ 24120573120588
31198882
+1212057321205742minus 16120574
31205881198882+ 24120588
31198882120574 minus 3120574120573
3))
times (4 (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882+ 312058821205733
+ 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882minus 312057331205742minus 312057431205732))
minus1
minus
(41205881198882120573 + 4119888
2120588120574 minus 3120573
2)
4120574 (120583 + 120573)
tanh2radicminus120573 (2120574 + 120573)
2120572 (2120574 + 120573)
times(119909 minus
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
119905)
1199081(119909 119905) = (120573 (120588
212057321198882+ 120574211988821205732+ 12057311988821205743+ 2120574212058821198882
+312057412058821205731198882minus
3
2
1205881205732120574 minus
3
4
1205881205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882+ 312058821205733
+ 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882minus 312057331205742minus 312057431205732)
minus1
plusmn (3 (minus 120573120588 + (minus12058821205732+ 41205743120573 minus 4120574
21198814
minus 412057431198812+ 212057321205742minus 41205882120574120573
+412058821198814+ 412058821198812120574)
12
))
times (4 (1205742minus 1205882))
minus1
times tanh2radicminus120573 (2120574 + 120573)
2120572 (2120574 + 120573)
times(119909 minus
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
119905)
1199062(119909 119905) = (120573 (6120588119888
21205732120574 + 10120573120574
21205881198882+
3
2
12058821205732+ 2120573120588
31198882
+312057321205742+ 412057431205881198882+ 212058831198882120574 +
9
4
1205741205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882minus 312058821205733
minus 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882+ 312057331205742+ 312057431205732)
minus1
minus
(41205881198882120573 + 4119888
2120588120574 + 3120573
2)
4120574 (120583 + 120573)
tanh2radic120573 (2120574 + 120573)
2120572 (2120574 + 120573)
times(119909 minus
radic120573 (412057221205832+ 1)
412057221205832+ 1
119905)
1199082(119909 119905) = (120573 (minus 3120588
212057321198882minus 3120574211988821205732minus 312057311988821205743
minus6120574212058821198882minus 9120574120588
21205731198882minus
9
2
1205881205732120574 minus
9
4
1205881205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882minus 312058821205733
minus 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882+ 312057331205742+ 312057431205732)
minus1
plusmn (3 (120573120588 + (minus12058821205732+ 41205743120573 minus 4120574
21198814
minus 412057431198812+ 212057321205742minus 41205882120574120573
+412058821198814+ 412058821198812120574)
12
))
times (4 (1205742minus 1205882))
minus1
times tanh2radic120573 (2120574 + 120573)
2120572 (2120574 + 120573)
(119909 minus
radic120573 (412057221205832+ 1)
412057221205832+ 1
119905)
(47)
4 Conclusion
In summary the tanh-coth method has been successfullyimplemented to find new travelling wave solutions for severalkinds of two coupled improved Boussinesq equations Thesolutions that have been found can be seen as an expansionand verification of the previously known data However thisstudy confirmed again the belief that the tanh-coth methodis a powerful and reliable technique to handle nonlineardispersive-dissipative equations and system of equationsThroughout the work Maple has been used to overcome thetedious algebraic calculations
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] A De Godefroy ldquoBlow up of solutions of a generalized Boussi-nesq equationrdquo IMA Journal of AppliedMathematics vol 60 no2 pp 123ndash138 1998
14 Journal of Applied Mathematics
[2] S Wang and M Li ldquoThe Cauchy problem for coupled IMBqequationsrdquo IMA Journal of Applied Mathematics vol 74 no 5pp 726ndash740 2009
[3] N Duruk H A Erbay and A Erkip ldquoBlow-up and globalexistence for a general class of nonlocal nonlinear coupled waveequationsrdquo Journal of Differential Equations vol 250 no 3 pp1448ndash1459 2011
[4] G Chen and H Zhang ldquoInitial boundary value problem for asystem of generalized IMBq equationsrdquoMathematical Methodsin the Applied Sciences vol 27 no 5 pp 497ndash518 2004
[5] P Rosenau ldquoDynamics of dense latticesrdquo Physical Review B vol36 no 11 pp 5868ndash5876 1987
[6] P L Christiansen P S Lomdahl andVMuto ldquoOn aToda latticemodel with a transversal degree of freedomrdquo Nonlinearity vol4 no 2 pp 477ndash501 1991
[7] S K Turitsyn ldquoOn a Toda lattice model with a transversaldegree of freedom Sufficient criterion of blow-up in thecontinuum limitrdquo Physics Letters A vol 173 no 3 pp 267ndash2691993
[8] R L Pego P Smereka and M I Weinstein ldquoOscillatoryinstability of solitary waves in a continuum model of latticevibrationsrdquo Nonlinearity vol 8 no 6 pp 921ndash941 1995
[9] X Fan and L Tian ldquoCompacton solutions and multiple com-pacton solutions for a continuum Toda lattice modelrdquo ChaosSolitons and Fractals vol 29 no 4 pp 882ndash894 2006
[10] A M Wazwaz ldquoThe Hirotarsquos bilinear method and the tanh-coth method for multiple-soliton solutions of the Sawada-Kotera-Kadomtsev-Petviashvili equationrdquoAppliedMathematicsand Computation vol 200 no 1 pp 160ndash166 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Journal of Applied Mathematics
Collecting the coefficients of 119884 and solving the resultingsystem we find the following sets of solutions
1198860=
120573 (3120573 + 8)
8 (120573 + 1)
1198862=
minus31205732
16 (120573 + 1)
1198872= 1198862 119888
0= minus2119889
2 119888
2= 1198892
1198892=
plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573
16
119881 =
radic120573 (1612057221205832+ 1)
1612057221205832+ 1
120583 =
radic120573 (120573 + 2)
4120572 (120573 + 2)
1198861= 1198871= 1198881= 1198891= 0
1198860=
120573 (120573 + 8)
8 (120573 + 1)
1198862=
31205732
16 (120573 + 1)
1198872= 1198862
1198880=
21198892
3
1198882= 1198892
1198892=
plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573
16
119881 =
radicminus120573 (1612057221205832minus 1)
1612057221205832minus 1
120583 =
radicminus120573 (120573 + 2)
4120572 (120573 + 2)
1198861= 1198871= 1198881= 1198891= 0
1198860=
minus120573 (120573 minus 4)
4 (120573 + 1)
1198862=
31205732
4 (120573 + 1)
1198880= minus
1198882
3
1198882=
plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573
4
119881 =
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
120583 =
radicminus120573 (120573 + 2)
2120572 (120573 + 2)
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860=
120573 (3120573 + 4)
4 (120573 + 1)
1198862=
minus31205732
4 (120573 + 1)
1198880= minus2
1198882=
plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573
4
119881 =
radic120573 (412057221205832+ 1)
412057221205832+ 1
120583 =
radic120573 (120573 + 2)
2120572 (120573 + 2)
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860=
120573
120573 + 1
1198862=
31205732
2 (31205732+ 7120573 + 4)
1198872= 1198862 119888
2= minus1198892
1198892=
plusmn3radicminus421198814120573 minus 58119881
4minus 21198812120573 minus 120119881
2+ 120120573
8
119881 =
radic5120573 (812057221205832+ 3120573 + 5)
812057221205832+ 3120573 + 5
120583 =
radicminus120573 (5120573 + 4)
2120572 (5120573 + 4)
1198861= 1198871= 1198880= 1198881= 1198891= 0
1198860=
120573
120573 + 1
1198862=
minus120573
2 (120573 + 1)
1198872= 1198862
1198882= minus1198892 119889
2=
plusmnradic91198812+ 6
2
119881 =
radic3120573 (120573 + 1)
3 (120573 + 1)
120583 =
minus1
2120572
1198861= 1198871= 1198880= 1198881= 1198891= 0
(32)
Consequently we obtain the following solutions
1199061(119909 119905) =
120573 (3120573 + 8)
8 (120573 + 1)
minus
31205732
16 (120573 + 1)
tanh2radic120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radic120573 (1612057221205832+ 1)
1612057221205832+ 1
119905)
minus
31205732
16 (120573 + 1)
coth2radic120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radic120573 (1612057221205832+ 1)
1612057221205832+ 1
119905)
1199081(119909 119905) =
∓3radicminus41198814minus 41198812+ 21205732+ 4120573
8
plusmn
3radicminus41198814minus 41198812+ 21205732+ 4120573
16
tanh2radic120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radic120573 (1612057221205832+ 1)
1612057221205832+ 1
119905)
plusmn
3radicminus41198814minus 41198812+ 21205732+ 4120573
16
coth2radic120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radic120573 (1612057221205832+ 1)
1612057221205832+ 1
119905)
Journal of Applied Mathematics 9
1199062(119909 119905) =
120573 (120573 + 8)
8 (120573 + 1)
+
31205732
16 (120573 + 1)
tanh2radicminus120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radicminus120573 (1612057221205832minus 1)
1612057221205832minus 1
119905)
+
31205732
16 (120573 + 1)
coth2radicminus120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radicminus120573 (1612057221205832minus 1)
1612057221205832minus 1
119905)
1199082(119909 119905) = plusmn
radicminus41198814minus 41198812+ 21205732+ 4120573
8
plusmn
3radicminus41198814minus 41198812+ 21205732+ 4120573
16
tanh2radicminus120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radicminus120573 (1612057221205832minus 1)
1612057221205832minus 1
119905)
plusmn
3radicminus41198814minus 41198812+ 21205732+ 4120573
16
coth2radicminus120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radicminus120573 (1612057221205832minus 1)
1612057221205832minus 1
119905)
1199063(119909 119905) =
minus120573 (120573 minus 4)
4 (120573 + 1)
+
31205732
4 (120573 + 1)
tanh2radicminus120573 (120573 + 2)
2120572 (120573 + 2)
times (119909 minus
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
119905)
1199083(119909 119905) = ∓
radicminus41198814minus 41198812+ 21205732+ 4120573
4
plusmn
3radicminus41198814minus 41198812+ 21205732+ 4120573
4
tanhradicminus120573 (120573 + 2)
2120572 (120573 + 2)
times (119909 minus
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
119905)
1199064(119909 119905) =
120573 (3120573 + 4)
4 (120573 + 1)
minus
31205732
4 (120573 + 1)
tanh2radic120573 (120573 + 2)
2120572 (120573 + 2)
times (119909 minus
radic120573 (412057221205832+ 1)
412057221205832+ 1
119905)
1199084(119909 119905) = ∓
3radicminus41198814minus 41198812+ 21205732+ 4120573
4
plusmn
3radicminus41198814minus 41198812+ 21205732+ 4120573
4
tanh2radic120573 (120573 + 2)
2120572 (120573 + 2)
times (119909 minus
radic120573 (412057221205832+ 1)
412057221205832+ 1
119905)
1199065(119909 119905) =
120573
120573 + 1
+
31205732
2 (31205732+ 7120573 + 4)
tanh2radicminus120573 (5120573 + 4)
2120572 (5120573 + 4)
times (119909 minus
radic5120573 (812057221205832+ 3120573 + 5)
812057221205832+ 3120573 + 5
119905)
+
31205732
2 (31205732+ 7120573 + 4)
coth2radicminus120573 (5120573 + 4)
2120572 (5120573 + 4)
times (119909 minus
radic5120573 (812057221205832+ 3120573 + 5)
812057221205832+ 3120573 + 5
119905)
1199085(119909 119905)
= ∓
3radicminus421198814120573 minus 58119881
4minus 21198812120573 minus 120119881
2+ 120120573
8
times tanh2radicminus120573 (5120573 + 4)
2120572 (5120573 + 4)
times (119909 minus
radic5120573 (812057221205832+ 3120573 + 5)
812057221205832+ 3120573 + 5
119905)
plusmn
3radicminus421198814120573 minus 58119881
4minus 21198812120573 minus 120119881
2+ 120120573
8
10 Journal of Applied Mathematics
times coth2radicminus120573 (5120573 + 4)
2120572 (5120573 + 4)
times (119909 minus
radic5120573 (812057221205832+ 3120573 + 5)
812057221205832+ 3120573 + 5
119905)
1199066(119909 119905) =
120573
120573 + 1
minus
120573
2 (120573 + 1)
tanh2minus12120572
(119909 minus
radic3120573 (120573 + 1)
3 (120573 + 1)
119905)
minus
120573
2 (120573 + 1)
coth2minus12120572
(119909 minus
radic3120573 (120573 + 1)
3 (120573 + 1)
119905)
1199086(119909 119905)
= ∓
radic91198812+ 6
2
tanh2minus12120572
(119909 minus
radic3120573 (120573 + 1)
3 (120573 + 1)
119905)
plusmn
radic91198812+ 6
2
coth2minus12120572
(119909 minus
radic3120573 (120573 + 1)
3 (120573 + 1)
119905)
(33)
34 The Fourth Coupled We now consider the coupled IBqequations
119906119905119905= 119906119909119909minus 120573(119906
2)119909119909+ (1199082)119909119909+ 119906119909119909119905119905
119908119905119905= 119908119909119909+ 2(119906119908)
119909119909+ 119908119909119909119905119905
(34)
Using the wave variable 120585 = 119909 minus 119881119905 (34) can be converted toa system of ODEs
(1198812minus 1)119880 + 120573119880
2minus1198822minus 119881211988010158401015840= 0
(1198812minus 1)119882 minus 2119880119882 minus 119881
211988210158401015840= 0
(35)
Balancing1198822 with 11988010158401015840 and 119880119882 with11988210158401015840 in (35) gives
2119873 = 119872 + 2
119872 + 119873 = 119873 + 2
(36)
so that
119872 = 119873 = 2 (37)
The tanh-coth method substitutes the finite expansions
119880(120583120585) = 119878 (119884) =
2
sum
119896=0
119886119896119884119896+
2
sum
119896=1
119887119896119884minus119896
119882 (120583120585) = 119878 (119884) =
2
sum
119896=0
119888119896119884119896+
2
sum
119896=1
119889119896119884119896
(38)
In (35) collecting the coefficients of each power of 119884 andsolving the resulting system we obtain
1198860=
minus21205832
161205832+ 1
1198862=
minus31205832
161205832+ 1
1198862= 1198872
1198880=
21198892
3
1198882= 1198892
1198892=
plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
119881 =
1
radic161205832+ 1
1198861= 1198871= 1198881= 1198891= 0
1198860=
minus61205832
161205832minus 1
1198862=
31205832
161205832minus 1
1198862= 1198872
1198880= minus2119889
2 119888
2= 1198892
1198892=
plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
119881 =
1
radicminus161205832+ 1
1198861= 1198871= 1198881= 1198891= 0
1198860=
1205832
41205832+ 1
1198862=
minus31205832
41205832+ 1
1198880=
minus1198882
3
1198882=
plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
119881 =
1
radic41205832+ 1
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860=
minus31205832
41205832minus 1
1198862=
31205832
41205832minus 1
1198880= minus1198882
1198882=
plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
119881 =
1
radicminus41205832+ 1
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860=
minus31205832
41205832minus 1
1198872=
31205832
41205832minus 1
1198880= minus1198892
1198892=
plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
119881 =
1
radicminus41205832+ 1
1198861= 1198862= 1198871= 1198881= 1198882= 1198891= 0
(39)
Journal of Applied Mathematics 11
where 120583 is left as a free parameter Consequently we obtainthe following travelling wave solutions
1199061(119909 119905) = minus
21205832
161205832+ 1
minus
31205832
161205832+ 1
tanh2120583(119909 minus 1
radic161205832+ 1
119905)
minus
31205832
161205832+ 1
coth2120583(119909 minus 1
radic161205832+ 1
119905)
1199081(119909 119905) = plusmn
(119881 + 1) (119881 minus 1)radic120573 + 2
8
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
tanh2120583
times(119909 minus
1
radic161205832+ 1
119905)
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
coth2120583
times(119909 minus
1
radic161205832+ 1
119905)
1199062(119909 119905) = minus
61205832
161205832minus 1
+
31205832
161205832minus 1
tanh2120583(119909 minus 1
radicminus161205832+ 1
119905)
+
31205832
161205832minus 1
coth2120583(119909 minus 1
radicminus161205832+ 1
119905)
1199082(119909 119905) = ∓
3 (119881 + 1) (119881 minus 1)radic120573 + 2
8
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
tanh2120583
times(119909 minus
1
radicminus161205832+ 1
119905)
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
coth2120583
times(119909 minus
1
radicminus161205832+ 1
119905)
1199063(119909 119905) =
1205832
41205832+ 1
minus
31205832
41205832+ 1
tanh2120583(119909 minus 1
radic41205832+ 1
119905)
1199083(119909 119905) = ∓
(119881 + 1) (119881 minus 1)radic120573 + 2
4
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
times tanh2120583(119909 minus 1
radic41205832+ 1
119905)
1199064(119909 119905) = minus
31205832
41205832minus 1
+
31205832
41205832minus 1
tanh2120583(119909 minus 1
radicminus41205832+ 1
119905)
1199084(119909 119905) = ∓
3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
times tanh2120583(119909 minus 1
radicminus41205832+ 1
119905)
1199065(119909 119905) = minus
31205832
41205832minus 1
+
31205832
41205832minus 1
coth2120583(119909 minus 1
radicminus41205832+ 1
119905)
1199085(119909 119905) = ∓
3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
plusmn 3
(119881 + 1) (119881 minus 1)radic120573 + 2
4
times coth2120583(119909 minus 1
radicminus41205832+ 1
119905)
(40)
35 The Fifth Coupled We finally consider the coupled IBqequations
119906119905119905= 120573119906119909119909minus
120573
2
(1199062)119909119909+
120573
2
(1199082)119909119909+ 1205722119906119909119909119905119905
119908119905119905= 120574(119906119908)
119909119909+ 120588(119908
2)119909119909+ 1205722119908119909119909119905119905
(41)
12 Journal of Applied Mathematics
This system of equations (41) can be converted to the follow-ing system of ODEs by using the wave variable 120585 = 119909 minus 119881119905
(21198812minus 2120573)119880 + 120573119880
2minus 120573119882
2minus 21205722119881211988010158401015840= 0
1198812119882minus 120574119880119882 minus 120588119882
2minus 1205722119881211988210158401015840= 0
(42)
Using the balancing procedure in (42) gives
2119872 = 119872 +119873
2119873 = 119872 +119873
(43)
so that
119872 = 119873 = 2 (44)
We consider solutions in the following form
119880 (120583120585) = 119878 (119884) =
2
sum
119896=0
119886119896119884119896+
2
sum
119896=1
119887119896119884minus119896
119882 (120583120585) = 119878 (119884) =
2
sum
119896=0
119888119896119884119896+
2
sum
119896=1
119889119896119884119896
(45)
Substituting (45) into (42) collecting the coefficients of 119884and solving the resulting system we find the following setsof solutions
1198860= (minus120573 (8120588119888
21205732120574 minus 8120573120574
21205881198882minus 18120588
21205732+ 24120573120588
31198882
+1212057321205742minus 16120574
31205881198882+ 24120588
31198882120574 minus 3120574120573
3))
times (4 (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882+ 312058821205733
+ 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882minus 312057331205742minus 312057431205732))
minus1
1198862=
minus (41205881198882120573 + 4119888
2120588120574 minus 3120573
2)
4120574 (120583 + 120573)
1198880= (120573(120588
212057321198882+ 120574211988821205732+ 12057311988821205743+ 2120574212058821198882
+312057412058821205731198882minus
3
2
1205881205732120574 minus
3
4
1205881205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882+ 312058821205733
+ 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882minus 312057331205742minus 312057431205732)
minus1
1198882= (plusmn3 (minus 120573120588 + (minus120588
21205732+ 41205743120573 minus 4120574
21198814
minus 412057431198812+ 212057321205742
minus41205882120574120573 + 4120588
21198814+ 412058821198812120574)
12
))
times (4 (1205742minus 1205882))
minus1
119881 =
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
120583 =
radicminus120573 (2120574 + 120573)
2120572 (2120574 + 120573)
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860= (120573(6120588119888
21205732120574 + 10120573120574
21205881198882+
3
2
12058821205732+ 2120573120588
31198882
+312057321205742+ 412057431205881198882+ 212058831198882120574 +
9
4
1205741205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882minus 312058821205733
minus 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882+ 312057331205742+ 312057431205732)
minus1
1198862=
minus (41205881198882120573 + 4119888
2120588120574 + 3120573
2)
4120574 (120583 + 120573)
1198880= (120573(minus 3120588
212057321198882minus 3120574211988821205732minus 312057311988821205743
minus6120574212058821198882minus 9120574120588
21205731198882minus
9
2
1205881205732120574 minus
9
4
1205881205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882
minus 312058821205733minus 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882+ 312057331205742+ 312057431205732)
minus1
1198882= (plusmn3 (120573120588 + (minus120588
21205732+ 41205743120573 minus 4120574
21198814
minus 412057431198812+ 212057321205742minus 41205882120574120573
+412058821198814+ 412058821198812120574)
12
))
times (4 (1205742minus 1205882))
minus1
119881 =
radic120573 (412057221205832+ 1)
412057221205832+ 1
120583 =
radic120573 (2120574 + 120573)
2120572 (2120574 + 120573)
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
(46)
Journal of Applied Mathematics 13
In view of this we obtain the following travelling wavesolutions
1199061(119909 119905) = (minus120573 (8120588119888
21205732120574 minus 8120573120574
21205881198882minus 18120588
21205732+ 24120573120588
31198882
+1212057321205742minus 16120574
31205881198882+ 24120588
31198882120574 minus 3120574120573
3))
times (4 (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882+ 312058821205733
+ 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882minus 312057331205742minus 312057431205732))
minus1
minus
(41205881198882120573 + 4119888
2120588120574 minus 3120573
2)
4120574 (120583 + 120573)
tanh2radicminus120573 (2120574 + 120573)
2120572 (2120574 + 120573)
times(119909 minus
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
119905)
1199081(119909 119905) = (120573 (120588
212057321198882+ 120574211988821205732+ 12057311988821205743+ 2120574212058821198882
+312057412058821205731198882minus
3
2
1205881205732120574 minus
3
4
1205881205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882+ 312058821205733
+ 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882minus 312057331205742minus 312057431205732)
minus1
plusmn (3 (minus 120573120588 + (minus12058821205732+ 41205743120573 minus 4120574
21198814
minus 412057431198812+ 212057321205742minus 41205882120574120573
+412058821198814+ 412058821198812120574)
12
))
times (4 (1205742minus 1205882))
minus1
times tanh2radicminus120573 (2120574 + 120573)
2120572 (2120574 + 120573)
times(119909 minus
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
119905)
1199062(119909 119905) = (120573 (6120588119888
21205732120574 + 10120573120574
21205881198882+
3
2
12058821205732+ 2120573120588
31198882
+312057321205742+ 412057431205881198882+ 212058831198882120574 +
9
4
1205741205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882minus 312058821205733
minus 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882+ 312057331205742+ 312057431205732)
minus1
minus
(41205881198882120573 + 4119888
2120588120574 + 3120573
2)
4120574 (120583 + 120573)
tanh2radic120573 (2120574 + 120573)
2120572 (2120574 + 120573)
times(119909 minus
radic120573 (412057221205832+ 1)
412057221205832+ 1
119905)
1199082(119909 119905) = (120573 (minus 3120588
212057321198882minus 3120574211988821205732minus 312057311988821205743
minus6120574212058821198882minus 9120574120588
21205731198882minus
9
2
1205881205732120574 minus
9
4
1205881205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882minus 312058821205733
minus 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882+ 312057331205742+ 312057431205732)
minus1
plusmn (3 (120573120588 + (minus12058821205732+ 41205743120573 minus 4120574
21198814
minus 412057431198812+ 212057321205742minus 41205882120574120573
+412058821198814+ 412058821198812120574)
12
))
times (4 (1205742minus 1205882))
minus1
times tanh2radic120573 (2120574 + 120573)
2120572 (2120574 + 120573)
(119909 minus
radic120573 (412057221205832+ 1)
412057221205832+ 1
119905)
(47)
4 Conclusion
In summary the tanh-coth method has been successfullyimplemented to find new travelling wave solutions for severalkinds of two coupled improved Boussinesq equations Thesolutions that have been found can be seen as an expansionand verification of the previously known data However thisstudy confirmed again the belief that the tanh-coth methodis a powerful and reliable technique to handle nonlineardispersive-dissipative equations and system of equationsThroughout the work Maple has been used to overcome thetedious algebraic calculations
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] A De Godefroy ldquoBlow up of solutions of a generalized Boussi-nesq equationrdquo IMA Journal of AppliedMathematics vol 60 no2 pp 123ndash138 1998
14 Journal of Applied Mathematics
[2] S Wang and M Li ldquoThe Cauchy problem for coupled IMBqequationsrdquo IMA Journal of Applied Mathematics vol 74 no 5pp 726ndash740 2009
[3] N Duruk H A Erbay and A Erkip ldquoBlow-up and globalexistence for a general class of nonlocal nonlinear coupled waveequationsrdquo Journal of Differential Equations vol 250 no 3 pp1448ndash1459 2011
[4] G Chen and H Zhang ldquoInitial boundary value problem for asystem of generalized IMBq equationsrdquoMathematical Methodsin the Applied Sciences vol 27 no 5 pp 497ndash518 2004
[5] P Rosenau ldquoDynamics of dense latticesrdquo Physical Review B vol36 no 11 pp 5868ndash5876 1987
[6] P L Christiansen P S Lomdahl andVMuto ldquoOn aToda latticemodel with a transversal degree of freedomrdquo Nonlinearity vol4 no 2 pp 477ndash501 1991
[7] S K Turitsyn ldquoOn a Toda lattice model with a transversaldegree of freedom Sufficient criterion of blow-up in thecontinuum limitrdquo Physics Letters A vol 173 no 3 pp 267ndash2691993
[8] R L Pego P Smereka and M I Weinstein ldquoOscillatoryinstability of solitary waves in a continuum model of latticevibrationsrdquo Nonlinearity vol 8 no 6 pp 921ndash941 1995
[9] X Fan and L Tian ldquoCompacton solutions and multiple com-pacton solutions for a continuum Toda lattice modelrdquo ChaosSolitons and Fractals vol 29 no 4 pp 882ndash894 2006
[10] A M Wazwaz ldquoThe Hirotarsquos bilinear method and the tanh-coth method for multiple-soliton solutions of the Sawada-Kotera-Kadomtsev-Petviashvili equationrdquoAppliedMathematicsand Computation vol 200 no 1 pp 160ndash166 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 9
1199062(119909 119905) =
120573 (120573 + 8)
8 (120573 + 1)
+
31205732
16 (120573 + 1)
tanh2radicminus120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radicminus120573 (1612057221205832minus 1)
1612057221205832minus 1
119905)
+
31205732
16 (120573 + 1)
coth2radicminus120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radicminus120573 (1612057221205832minus 1)
1612057221205832minus 1
119905)
1199082(119909 119905) = plusmn
radicminus41198814minus 41198812+ 21205732+ 4120573
8
plusmn
3radicminus41198814minus 41198812+ 21205732+ 4120573
16
tanh2radicminus120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radicminus120573 (1612057221205832minus 1)
1612057221205832minus 1
119905)
plusmn
3radicminus41198814minus 41198812+ 21205732+ 4120573
16
coth2radicminus120573 (120573 + 2)
4120572 (120573 + 2)
times (119909 minus
radicminus120573 (1612057221205832minus 1)
1612057221205832minus 1
119905)
1199063(119909 119905) =
minus120573 (120573 minus 4)
4 (120573 + 1)
+
31205732
4 (120573 + 1)
tanh2radicminus120573 (120573 + 2)
2120572 (120573 + 2)
times (119909 minus
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
119905)
1199083(119909 119905) = ∓
radicminus41198814minus 41198812+ 21205732+ 4120573
4
plusmn
3radicminus41198814minus 41198812+ 21205732+ 4120573
4
tanhradicminus120573 (120573 + 2)
2120572 (120573 + 2)
times (119909 minus
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
119905)
1199064(119909 119905) =
120573 (3120573 + 4)
4 (120573 + 1)
minus
31205732
4 (120573 + 1)
tanh2radic120573 (120573 + 2)
2120572 (120573 + 2)
times (119909 minus
radic120573 (412057221205832+ 1)
412057221205832+ 1
119905)
1199084(119909 119905) = ∓
3radicminus41198814minus 41198812+ 21205732+ 4120573
4
plusmn
3radicminus41198814minus 41198812+ 21205732+ 4120573
4
tanh2radic120573 (120573 + 2)
2120572 (120573 + 2)
times (119909 minus
radic120573 (412057221205832+ 1)
412057221205832+ 1
119905)
1199065(119909 119905) =
120573
120573 + 1
+
31205732
2 (31205732+ 7120573 + 4)
tanh2radicminus120573 (5120573 + 4)
2120572 (5120573 + 4)
times (119909 minus
radic5120573 (812057221205832+ 3120573 + 5)
812057221205832+ 3120573 + 5
119905)
+
31205732
2 (31205732+ 7120573 + 4)
coth2radicminus120573 (5120573 + 4)
2120572 (5120573 + 4)
times (119909 minus
radic5120573 (812057221205832+ 3120573 + 5)
812057221205832+ 3120573 + 5
119905)
1199085(119909 119905)
= ∓
3radicminus421198814120573 minus 58119881
4minus 21198812120573 minus 120119881
2+ 120120573
8
times tanh2radicminus120573 (5120573 + 4)
2120572 (5120573 + 4)
times (119909 minus
radic5120573 (812057221205832+ 3120573 + 5)
812057221205832+ 3120573 + 5
119905)
plusmn
3radicminus421198814120573 minus 58119881
4minus 21198812120573 minus 120119881
2+ 120120573
8
10 Journal of Applied Mathematics
times coth2radicminus120573 (5120573 + 4)
2120572 (5120573 + 4)
times (119909 minus
radic5120573 (812057221205832+ 3120573 + 5)
812057221205832+ 3120573 + 5
119905)
1199066(119909 119905) =
120573
120573 + 1
minus
120573
2 (120573 + 1)
tanh2minus12120572
(119909 minus
radic3120573 (120573 + 1)
3 (120573 + 1)
119905)
minus
120573
2 (120573 + 1)
coth2minus12120572
(119909 minus
radic3120573 (120573 + 1)
3 (120573 + 1)
119905)
1199086(119909 119905)
= ∓
radic91198812+ 6
2
tanh2minus12120572
(119909 minus
radic3120573 (120573 + 1)
3 (120573 + 1)
119905)
plusmn
radic91198812+ 6
2
coth2minus12120572
(119909 minus
radic3120573 (120573 + 1)
3 (120573 + 1)
119905)
(33)
34 The Fourth Coupled We now consider the coupled IBqequations
119906119905119905= 119906119909119909minus 120573(119906
2)119909119909+ (1199082)119909119909+ 119906119909119909119905119905
119908119905119905= 119908119909119909+ 2(119906119908)
119909119909+ 119908119909119909119905119905
(34)
Using the wave variable 120585 = 119909 minus 119881119905 (34) can be converted toa system of ODEs
(1198812minus 1)119880 + 120573119880
2minus1198822minus 119881211988010158401015840= 0
(1198812minus 1)119882 minus 2119880119882 minus 119881
211988210158401015840= 0
(35)
Balancing1198822 with 11988010158401015840 and 119880119882 with11988210158401015840 in (35) gives
2119873 = 119872 + 2
119872 + 119873 = 119873 + 2
(36)
so that
119872 = 119873 = 2 (37)
The tanh-coth method substitutes the finite expansions
119880(120583120585) = 119878 (119884) =
2
sum
119896=0
119886119896119884119896+
2
sum
119896=1
119887119896119884minus119896
119882 (120583120585) = 119878 (119884) =
2
sum
119896=0
119888119896119884119896+
2
sum
119896=1
119889119896119884119896
(38)
In (35) collecting the coefficients of each power of 119884 andsolving the resulting system we obtain
1198860=
minus21205832
161205832+ 1
1198862=
minus31205832
161205832+ 1
1198862= 1198872
1198880=
21198892
3
1198882= 1198892
1198892=
plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
119881 =
1
radic161205832+ 1
1198861= 1198871= 1198881= 1198891= 0
1198860=
minus61205832
161205832minus 1
1198862=
31205832
161205832minus 1
1198862= 1198872
1198880= minus2119889
2 119888
2= 1198892
1198892=
plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
119881 =
1
radicminus161205832+ 1
1198861= 1198871= 1198881= 1198891= 0
1198860=
1205832
41205832+ 1
1198862=
minus31205832
41205832+ 1
1198880=
minus1198882
3
1198882=
plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
119881 =
1
radic41205832+ 1
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860=
minus31205832
41205832minus 1
1198862=
31205832
41205832minus 1
1198880= minus1198882
1198882=
plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
119881 =
1
radicminus41205832+ 1
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860=
minus31205832
41205832minus 1
1198872=
31205832
41205832minus 1
1198880= minus1198892
1198892=
plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
119881 =
1
radicminus41205832+ 1
1198861= 1198862= 1198871= 1198881= 1198882= 1198891= 0
(39)
Journal of Applied Mathematics 11
where 120583 is left as a free parameter Consequently we obtainthe following travelling wave solutions
1199061(119909 119905) = minus
21205832
161205832+ 1
minus
31205832
161205832+ 1
tanh2120583(119909 minus 1
radic161205832+ 1
119905)
minus
31205832
161205832+ 1
coth2120583(119909 minus 1
radic161205832+ 1
119905)
1199081(119909 119905) = plusmn
(119881 + 1) (119881 minus 1)radic120573 + 2
8
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
tanh2120583
times(119909 minus
1
radic161205832+ 1
119905)
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
coth2120583
times(119909 minus
1
radic161205832+ 1
119905)
1199062(119909 119905) = minus
61205832
161205832minus 1
+
31205832
161205832minus 1
tanh2120583(119909 minus 1
radicminus161205832+ 1
119905)
+
31205832
161205832minus 1
coth2120583(119909 minus 1
radicminus161205832+ 1
119905)
1199082(119909 119905) = ∓
3 (119881 + 1) (119881 minus 1)radic120573 + 2
8
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
tanh2120583
times(119909 minus
1
radicminus161205832+ 1
119905)
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
coth2120583
times(119909 minus
1
radicminus161205832+ 1
119905)
1199063(119909 119905) =
1205832
41205832+ 1
minus
31205832
41205832+ 1
tanh2120583(119909 minus 1
radic41205832+ 1
119905)
1199083(119909 119905) = ∓
(119881 + 1) (119881 minus 1)radic120573 + 2
4
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
times tanh2120583(119909 minus 1
radic41205832+ 1
119905)
1199064(119909 119905) = minus
31205832
41205832minus 1
+
31205832
41205832minus 1
tanh2120583(119909 minus 1
radicminus41205832+ 1
119905)
1199084(119909 119905) = ∓
3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
times tanh2120583(119909 minus 1
radicminus41205832+ 1
119905)
1199065(119909 119905) = minus
31205832
41205832minus 1
+
31205832
41205832minus 1
coth2120583(119909 minus 1
radicminus41205832+ 1
119905)
1199085(119909 119905) = ∓
3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
plusmn 3
(119881 + 1) (119881 minus 1)radic120573 + 2
4
times coth2120583(119909 minus 1
radicminus41205832+ 1
119905)
(40)
35 The Fifth Coupled We finally consider the coupled IBqequations
119906119905119905= 120573119906119909119909minus
120573
2
(1199062)119909119909+
120573
2
(1199082)119909119909+ 1205722119906119909119909119905119905
119908119905119905= 120574(119906119908)
119909119909+ 120588(119908
2)119909119909+ 1205722119908119909119909119905119905
(41)
12 Journal of Applied Mathematics
This system of equations (41) can be converted to the follow-ing system of ODEs by using the wave variable 120585 = 119909 minus 119881119905
(21198812minus 2120573)119880 + 120573119880
2minus 120573119882
2minus 21205722119881211988010158401015840= 0
1198812119882minus 120574119880119882 minus 120588119882
2minus 1205722119881211988210158401015840= 0
(42)
Using the balancing procedure in (42) gives
2119872 = 119872 +119873
2119873 = 119872 +119873
(43)
so that
119872 = 119873 = 2 (44)
We consider solutions in the following form
119880 (120583120585) = 119878 (119884) =
2
sum
119896=0
119886119896119884119896+
2
sum
119896=1
119887119896119884minus119896
119882 (120583120585) = 119878 (119884) =
2
sum
119896=0
119888119896119884119896+
2
sum
119896=1
119889119896119884119896
(45)
Substituting (45) into (42) collecting the coefficients of 119884and solving the resulting system we find the following setsof solutions
1198860= (minus120573 (8120588119888
21205732120574 minus 8120573120574
21205881198882minus 18120588
21205732+ 24120573120588
31198882
+1212057321205742minus 16120574
31205881198882+ 24120588
31198882120574 minus 3120574120573
3))
times (4 (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882+ 312058821205733
+ 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882minus 312057331205742minus 312057431205732))
minus1
1198862=
minus (41205881198882120573 + 4119888
2120588120574 minus 3120573
2)
4120574 (120583 + 120573)
1198880= (120573(120588
212057321198882+ 120574211988821205732+ 12057311988821205743+ 2120574212058821198882
+312057412058821205731198882minus
3
2
1205881205732120574 minus
3
4
1205881205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882+ 312058821205733
+ 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882minus 312057331205742minus 312057431205732)
minus1
1198882= (plusmn3 (minus 120573120588 + (minus120588
21205732+ 41205743120573 minus 4120574
21198814
minus 412057431198812+ 212057321205742
minus41205882120574120573 + 4120588
21198814+ 412058821198812120574)
12
))
times (4 (1205742minus 1205882))
minus1
119881 =
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
120583 =
radicminus120573 (2120574 + 120573)
2120572 (2120574 + 120573)
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860= (120573(6120588119888
21205732120574 + 10120573120574
21205881198882+
3
2
12058821205732+ 2120573120588
31198882
+312057321205742+ 412057431205881198882+ 212058831198882120574 +
9
4
1205741205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882minus 312058821205733
minus 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882+ 312057331205742+ 312057431205732)
minus1
1198862=
minus (41205881198882120573 + 4119888
2120588120574 + 3120573
2)
4120574 (120583 + 120573)
1198880= (120573(minus 3120588
212057321198882minus 3120574211988821205732minus 312057311988821205743
minus6120574212058821198882minus 9120574120588
21205731198882minus
9
2
1205881205732120574 minus
9
4
1205881205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882
minus 312058821205733minus 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882+ 312057331205742+ 312057431205732)
minus1
1198882= (plusmn3 (120573120588 + (minus120588
21205732+ 41205743120573 minus 4120574
21198814
minus 412057431198812+ 212057321205742minus 41205882120574120573
+412058821198814+ 412058821198812120574)
12
))
times (4 (1205742minus 1205882))
minus1
119881 =
radic120573 (412057221205832+ 1)
412057221205832+ 1
120583 =
radic120573 (2120574 + 120573)
2120572 (2120574 + 120573)
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
(46)
Journal of Applied Mathematics 13
In view of this we obtain the following travelling wavesolutions
1199061(119909 119905) = (minus120573 (8120588119888
21205732120574 minus 8120573120574
21205881198882minus 18120588
21205732+ 24120573120588
31198882
+1212057321205742minus 16120574
31205881198882+ 24120588
31198882120574 minus 3120574120573
3))
times (4 (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882+ 312058821205733
+ 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882minus 312057331205742minus 312057431205732))
minus1
minus
(41205881198882120573 + 4119888
2120588120574 minus 3120573
2)
4120574 (120583 + 120573)
tanh2radicminus120573 (2120574 + 120573)
2120572 (2120574 + 120573)
times(119909 minus
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
119905)
1199081(119909 119905) = (120573 (120588
212057321198882+ 120574211988821205732+ 12057311988821205743+ 2120574212058821198882
+312057412058821205731198882minus
3
2
1205881205732120574 minus
3
4
1205881205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882+ 312058821205733
+ 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882minus 312057331205742minus 312057431205732)
minus1
plusmn (3 (minus 120573120588 + (minus12058821205732+ 41205743120573 minus 4120574
21198814
minus 412057431198812+ 212057321205742minus 41205882120574120573
+412058821198814+ 412058821198812120574)
12
))
times (4 (1205742minus 1205882))
minus1
times tanh2radicminus120573 (2120574 + 120573)
2120572 (2120574 + 120573)
times(119909 minus
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
119905)
1199062(119909 119905) = (120573 (6120588119888
21205732120574 + 10120573120574
21205881198882+
3
2
12058821205732+ 2120573120588
31198882
+312057321205742+ 412057431205881198882+ 212058831198882120574 +
9
4
1205741205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882minus 312058821205733
minus 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882+ 312057331205742+ 312057431205732)
minus1
minus
(41205881198882120573 + 4119888
2120588120574 + 3120573
2)
4120574 (120583 + 120573)
tanh2radic120573 (2120574 + 120573)
2120572 (2120574 + 120573)
times(119909 minus
radic120573 (412057221205832+ 1)
412057221205832+ 1
119905)
1199082(119909 119905) = (120573 (minus 3120588
212057321198882minus 3120574211988821205732minus 312057311988821205743
minus6120574212058821198882minus 9120574120588
21205731198882minus
9
2
1205881205732120574 minus
9
4
1205881205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882minus 312058821205733
minus 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882+ 312057331205742+ 312057431205732)
minus1
plusmn (3 (120573120588 + (minus12058821205732+ 41205743120573 minus 4120574
21198814
minus 412057431198812+ 212057321205742minus 41205882120574120573
+412058821198814+ 412058821198812120574)
12
))
times (4 (1205742minus 1205882))
minus1
times tanh2radic120573 (2120574 + 120573)
2120572 (2120574 + 120573)
(119909 minus
radic120573 (412057221205832+ 1)
412057221205832+ 1
119905)
(47)
4 Conclusion
In summary the tanh-coth method has been successfullyimplemented to find new travelling wave solutions for severalkinds of two coupled improved Boussinesq equations Thesolutions that have been found can be seen as an expansionand verification of the previously known data However thisstudy confirmed again the belief that the tanh-coth methodis a powerful and reliable technique to handle nonlineardispersive-dissipative equations and system of equationsThroughout the work Maple has been used to overcome thetedious algebraic calculations
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] A De Godefroy ldquoBlow up of solutions of a generalized Boussi-nesq equationrdquo IMA Journal of AppliedMathematics vol 60 no2 pp 123ndash138 1998
14 Journal of Applied Mathematics
[2] S Wang and M Li ldquoThe Cauchy problem for coupled IMBqequationsrdquo IMA Journal of Applied Mathematics vol 74 no 5pp 726ndash740 2009
[3] N Duruk H A Erbay and A Erkip ldquoBlow-up and globalexistence for a general class of nonlocal nonlinear coupled waveequationsrdquo Journal of Differential Equations vol 250 no 3 pp1448ndash1459 2011
[4] G Chen and H Zhang ldquoInitial boundary value problem for asystem of generalized IMBq equationsrdquoMathematical Methodsin the Applied Sciences vol 27 no 5 pp 497ndash518 2004
[5] P Rosenau ldquoDynamics of dense latticesrdquo Physical Review B vol36 no 11 pp 5868ndash5876 1987
[6] P L Christiansen P S Lomdahl andVMuto ldquoOn aToda latticemodel with a transversal degree of freedomrdquo Nonlinearity vol4 no 2 pp 477ndash501 1991
[7] S K Turitsyn ldquoOn a Toda lattice model with a transversaldegree of freedom Sufficient criterion of blow-up in thecontinuum limitrdquo Physics Letters A vol 173 no 3 pp 267ndash2691993
[8] R L Pego P Smereka and M I Weinstein ldquoOscillatoryinstability of solitary waves in a continuum model of latticevibrationsrdquo Nonlinearity vol 8 no 6 pp 921ndash941 1995
[9] X Fan and L Tian ldquoCompacton solutions and multiple com-pacton solutions for a continuum Toda lattice modelrdquo ChaosSolitons and Fractals vol 29 no 4 pp 882ndash894 2006
[10] A M Wazwaz ldquoThe Hirotarsquos bilinear method and the tanh-coth method for multiple-soliton solutions of the Sawada-Kotera-Kadomtsev-Petviashvili equationrdquoAppliedMathematicsand Computation vol 200 no 1 pp 160ndash166 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Journal of Applied Mathematics
times coth2radicminus120573 (5120573 + 4)
2120572 (5120573 + 4)
times (119909 minus
radic5120573 (812057221205832+ 3120573 + 5)
812057221205832+ 3120573 + 5
119905)
1199066(119909 119905) =
120573
120573 + 1
minus
120573
2 (120573 + 1)
tanh2minus12120572
(119909 minus
radic3120573 (120573 + 1)
3 (120573 + 1)
119905)
minus
120573
2 (120573 + 1)
coth2minus12120572
(119909 minus
radic3120573 (120573 + 1)
3 (120573 + 1)
119905)
1199086(119909 119905)
= ∓
radic91198812+ 6
2
tanh2minus12120572
(119909 minus
radic3120573 (120573 + 1)
3 (120573 + 1)
119905)
plusmn
radic91198812+ 6
2
coth2minus12120572
(119909 minus
radic3120573 (120573 + 1)
3 (120573 + 1)
119905)
(33)
34 The Fourth Coupled We now consider the coupled IBqequations
119906119905119905= 119906119909119909minus 120573(119906
2)119909119909+ (1199082)119909119909+ 119906119909119909119905119905
119908119905119905= 119908119909119909+ 2(119906119908)
119909119909+ 119908119909119909119905119905
(34)
Using the wave variable 120585 = 119909 minus 119881119905 (34) can be converted toa system of ODEs
(1198812minus 1)119880 + 120573119880
2minus1198822minus 119881211988010158401015840= 0
(1198812minus 1)119882 minus 2119880119882 minus 119881
211988210158401015840= 0
(35)
Balancing1198822 with 11988010158401015840 and 119880119882 with11988210158401015840 in (35) gives
2119873 = 119872 + 2
119872 + 119873 = 119873 + 2
(36)
so that
119872 = 119873 = 2 (37)
The tanh-coth method substitutes the finite expansions
119880(120583120585) = 119878 (119884) =
2
sum
119896=0
119886119896119884119896+
2
sum
119896=1
119887119896119884minus119896
119882 (120583120585) = 119878 (119884) =
2
sum
119896=0
119888119896119884119896+
2
sum
119896=1
119889119896119884119896
(38)
In (35) collecting the coefficients of each power of 119884 andsolving the resulting system we obtain
1198860=
minus21205832
161205832+ 1
1198862=
minus31205832
161205832+ 1
1198862= 1198872
1198880=
21198892
3
1198882= 1198892
1198892=
plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
119881 =
1
radic161205832+ 1
1198861= 1198871= 1198881= 1198891= 0
1198860=
minus61205832
161205832minus 1
1198862=
31205832
161205832minus 1
1198862= 1198872
1198880= minus2119889
2 119888
2= 1198892
1198892=
plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
119881 =
1
radicminus161205832+ 1
1198861= 1198871= 1198881= 1198891= 0
1198860=
1205832
41205832+ 1
1198862=
minus31205832
41205832+ 1
1198880=
minus1198882
3
1198882=
plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
119881 =
1
radic41205832+ 1
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860=
minus31205832
41205832minus 1
1198862=
31205832
41205832minus 1
1198880= minus1198882
1198882=
plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
119881 =
1
radicminus41205832+ 1
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860=
minus31205832
41205832minus 1
1198872=
31205832
41205832minus 1
1198880= minus1198892
1198892=
plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
119881 =
1
radicminus41205832+ 1
1198861= 1198862= 1198871= 1198881= 1198882= 1198891= 0
(39)
Journal of Applied Mathematics 11
where 120583 is left as a free parameter Consequently we obtainthe following travelling wave solutions
1199061(119909 119905) = minus
21205832
161205832+ 1
minus
31205832
161205832+ 1
tanh2120583(119909 minus 1
radic161205832+ 1
119905)
minus
31205832
161205832+ 1
coth2120583(119909 minus 1
radic161205832+ 1
119905)
1199081(119909 119905) = plusmn
(119881 + 1) (119881 minus 1)radic120573 + 2
8
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
tanh2120583
times(119909 minus
1
radic161205832+ 1
119905)
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
coth2120583
times(119909 minus
1
radic161205832+ 1
119905)
1199062(119909 119905) = minus
61205832
161205832minus 1
+
31205832
161205832minus 1
tanh2120583(119909 minus 1
radicminus161205832+ 1
119905)
+
31205832
161205832minus 1
coth2120583(119909 minus 1
radicminus161205832+ 1
119905)
1199082(119909 119905) = ∓
3 (119881 + 1) (119881 minus 1)radic120573 + 2
8
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
tanh2120583
times(119909 minus
1
radicminus161205832+ 1
119905)
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
coth2120583
times(119909 minus
1
radicminus161205832+ 1
119905)
1199063(119909 119905) =
1205832
41205832+ 1
minus
31205832
41205832+ 1
tanh2120583(119909 minus 1
radic41205832+ 1
119905)
1199083(119909 119905) = ∓
(119881 + 1) (119881 minus 1)radic120573 + 2
4
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
times tanh2120583(119909 minus 1
radic41205832+ 1
119905)
1199064(119909 119905) = minus
31205832
41205832minus 1
+
31205832
41205832minus 1
tanh2120583(119909 minus 1
radicminus41205832+ 1
119905)
1199084(119909 119905) = ∓
3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
times tanh2120583(119909 minus 1
radicminus41205832+ 1
119905)
1199065(119909 119905) = minus
31205832
41205832minus 1
+
31205832
41205832minus 1
coth2120583(119909 minus 1
radicminus41205832+ 1
119905)
1199085(119909 119905) = ∓
3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
plusmn 3
(119881 + 1) (119881 minus 1)radic120573 + 2
4
times coth2120583(119909 minus 1
radicminus41205832+ 1
119905)
(40)
35 The Fifth Coupled We finally consider the coupled IBqequations
119906119905119905= 120573119906119909119909minus
120573
2
(1199062)119909119909+
120573
2
(1199082)119909119909+ 1205722119906119909119909119905119905
119908119905119905= 120574(119906119908)
119909119909+ 120588(119908
2)119909119909+ 1205722119908119909119909119905119905
(41)
12 Journal of Applied Mathematics
This system of equations (41) can be converted to the follow-ing system of ODEs by using the wave variable 120585 = 119909 minus 119881119905
(21198812minus 2120573)119880 + 120573119880
2minus 120573119882
2minus 21205722119881211988010158401015840= 0
1198812119882minus 120574119880119882 minus 120588119882
2minus 1205722119881211988210158401015840= 0
(42)
Using the balancing procedure in (42) gives
2119872 = 119872 +119873
2119873 = 119872 +119873
(43)
so that
119872 = 119873 = 2 (44)
We consider solutions in the following form
119880 (120583120585) = 119878 (119884) =
2
sum
119896=0
119886119896119884119896+
2
sum
119896=1
119887119896119884minus119896
119882 (120583120585) = 119878 (119884) =
2
sum
119896=0
119888119896119884119896+
2
sum
119896=1
119889119896119884119896
(45)
Substituting (45) into (42) collecting the coefficients of 119884and solving the resulting system we find the following setsof solutions
1198860= (minus120573 (8120588119888
21205732120574 minus 8120573120574
21205881198882minus 18120588
21205732+ 24120573120588
31198882
+1212057321205742minus 16120574
31205881198882+ 24120588
31198882120574 minus 3120574120573
3))
times (4 (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882+ 312058821205733
+ 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882minus 312057331205742minus 312057431205732))
minus1
1198862=
minus (41205881198882120573 + 4119888
2120588120574 minus 3120573
2)
4120574 (120583 + 120573)
1198880= (120573(120588
212057321198882+ 120574211988821205732+ 12057311988821205743+ 2120574212058821198882
+312057412058821205731198882minus
3
2
1205881205732120574 minus
3
4
1205881205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882+ 312058821205733
+ 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882minus 312057331205742minus 312057431205732)
minus1
1198882= (plusmn3 (minus 120573120588 + (minus120588
21205732+ 41205743120573 minus 4120574
21198814
minus 412057431198812+ 212057321205742
minus41205882120574120573 + 4120588
21198814+ 412058821198812120574)
12
))
times (4 (1205742minus 1205882))
minus1
119881 =
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
120583 =
radicminus120573 (2120574 + 120573)
2120572 (2120574 + 120573)
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860= (120573(6120588119888
21205732120574 + 10120573120574
21205881198882+
3
2
12058821205732+ 2120573120588
31198882
+312057321205742+ 412057431205881198882+ 212058831198882120574 +
9
4
1205741205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882minus 312058821205733
minus 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882+ 312057331205742+ 312057431205732)
minus1
1198862=
minus (41205881198882120573 + 4119888
2120588120574 + 3120573
2)
4120574 (120583 + 120573)
1198880= (120573(minus 3120588
212057321198882minus 3120574211988821205732minus 312057311988821205743
minus6120574212058821198882minus 9120574120588
21205731198882minus
9
2
1205881205732120574 minus
9
4
1205881205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882
minus 312058821205733minus 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882+ 312057331205742+ 312057431205732)
minus1
1198882= (plusmn3 (120573120588 + (minus120588
21205732+ 41205743120573 minus 4120574
21198814
minus 412057431198812+ 212057321205742minus 41205882120574120573
+412058821198814+ 412058821198812120574)
12
))
times (4 (1205742minus 1205882))
minus1
119881 =
radic120573 (412057221205832+ 1)
412057221205832+ 1
120583 =
radic120573 (2120574 + 120573)
2120572 (2120574 + 120573)
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
(46)
Journal of Applied Mathematics 13
In view of this we obtain the following travelling wavesolutions
1199061(119909 119905) = (minus120573 (8120588119888
21205732120574 minus 8120573120574
21205881198882minus 18120588
21205732+ 24120573120588
31198882
+1212057321205742minus 16120574
31205881198882+ 24120588
31198882120574 minus 3120574120573
3))
times (4 (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882+ 312058821205733
+ 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882minus 312057331205742minus 312057431205732))
minus1
minus
(41205881198882120573 + 4119888
2120588120574 minus 3120573
2)
4120574 (120583 + 120573)
tanh2radicminus120573 (2120574 + 120573)
2120572 (2120574 + 120573)
times(119909 minus
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
119905)
1199081(119909 119905) = (120573 (120588
212057321198882+ 120574211988821205732+ 12057311988821205743+ 2120574212058821198882
+312057412058821205731198882minus
3
2
1205881205732120574 minus
3
4
1205881205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882+ 312058821205733
+ 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882minus 312057331205742minus 312057431205732)
minus1
plusmn (3 (minus 120573120588 + (minus12058821205732+ 41205743120573 minus 4120574
21198814
minus 412057431198812+ 212057321205742minus 41205882120574120573
+412058821198814+ 412058821198812120574)
12
))
times (4 (1205742minus 1205882))
minus1
times tanh2radicminus120573 (2120574 + 120573)
2120572 (2120574 + 120573)
times(119909 minus
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
119905)
1199062(119909 119905) = (120573 (6120588119888
21205732120574 + 10120573120574
21205881198882+
3
2
12058821205732+ 2120573120588
31198882
+312057321205742+ 412057431205881198882+ 212058831198882120574 +
9
4
1205741205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882minus 312058821205733
minus 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882+ 312057331205742+ 312057431205732)
minus1
minus
(41205881198882120573 + 4119888
2120588120574 + 3120573
2)
4120574 (120583 + 120573)
tanh2radic120573 (2120574 + 120573)
2120572 (2120574 + 120573)
times(119909 minus
radic120573 (412057221205832+ 1)
412057221205832+ 1
119905)
1199082(119909 119905) = (120573 (minus 3120588
212057321198882minus 3120574211988821205732minus 312057311988821205743
minus6120574212058821198882minus 9120574120588
21205731198882minus
9
2
1205881205732120574 minus
9
4
1205881205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882minus 312058821205733
minus 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882+ 312057331205742+ 312057431205732)
minus1
plusmn (3 (120573120588 + (minus12058821205732+ 41205743120573 minus 4120574
21198814
minus 412057431198812+ 212057321205742minus 41205882120574120573
+412058821198814+ 412058821198812120574)
12
))
times (4 (1205742minus 1205882))
minus1
times tanh2radic120573 (2120574 + 120573)
2120572 (2120574 + 120573)
(119909 minus
radic120573 (412057221205832+ 1)
412057221205832+ 1
119905)
(47)
4 Conclusion
In summary the tanh-coth method has been successfullyimplemented to find new travelling wave solutions for severalkinds of two coupled improved Boussinesq equations Thesolutions that have been found can be seen as an expansionand verification of the previously known data However thisstudy confirmed again the belief that the tanh-coth methodis a powerful and reliable technique to handle nonlineardispersive-dissipative equations and system of equationsThroughout the work Maple has been used to overcome thetedious algebraic calculations
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] A De Godefroy ldquoBlow up of solutions of a generalized Boussi-nesq equationrdquo IMA Journal of AppliedMathematics vol 60 no2 pp 123ndash138 1998
14 Journal of Applied Mathematics
[2] S Wang and M Li ldquoThe Cauchy problem for coupled IMBqequationsrdquo IMA Journal of Applied Mathematics vol 74 no 5pp 726ndash740 2009
[3] N Duruk H A Erbay and A Erkip ldquoBlow-up and globalexistence for a general class of nonlocal nonlinear coupled waveequationsrdquo Journal of Differential Equations vol 250 no 3 pp1448ndash1459 2011
[4] G Chen and H Zhang ldquoInitial boundary value problem for asystem of generalized IMBq equationsrdquoMathematical Methodsin the Applied Sciences vol 27 no 5 pp 497ndash518 2004
[5] P Rosenau ldquoDynamics of dense latticesrdquo Physical Review B vol36 no 11 pp 5868ndash5876 1987
[6] P L Christiansen P S Lomdahl andVMuto ldquoOn aToda latticemodel with a transversal degree of freedomrdquo Nonlinearity vol4 no 2 pp 477ndash501 1991
[7] S K Turitsyn ldquoOn a Toda lattice model with a transversaldegree of freedom Sufficient criterion of blow-up in thecontinuum limitrdquo Physics Letters A vol 173 no 3 pp 267ndash2691993
[8] R L Pego P Smereka and M I Weinstein ldquoOscillatoryinstability of solitary waves in a continuum model of latticevibrationsrdquo Nonlinearity vol 8 no 6 pp 921ndash941 1995
[9] X Fan and L Tian ldquoCompacton solutions and multiple com-pacton solutions for a continuum Toda lattice modelrdquo ChaosSolitons and Fractals vol 29 no 4 pp 882ndash894 2006
[10] A M Wazwaz ldquoThe Hirotarsquos bilinear method and the tanh-coth method for multiple-soliton solutions of the Sawada-Kotera-Kadomtsev-Petviashvili equationrdquoAppliedMathematicsand Computation vol 200 no 1 pp 160ndash166 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 11
where 120583 is left as a free parameter Consequently we obtainthe following travelling wave solutions
1199061(119909 119905) = minus
21205832
161205832+ 1
minus
31205832
161205832+ 1
tanh2120583(119909 minus 1
radic161205832+ 1
119905)
minus
31205832
161205832+ 1
coth2120583(119909 minus 1
radic161205832+ 1
119905)
1199081(119909 119905) = plusmn
(119881 + 1) (119881 minus 1)radic120573 + 2
8
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
tanh2120583
times(119909 minus
1
radic161205832+ 1
119905)
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
coth2120583
times(119909 minus
1
radic161205832+ 1
119905)
1199062(119909 119905) = minus
61205832
161205832minus 1
+
31205832
161205832minus 1
tanh2120583(119909 minus 1
radicminus161205832+ 1
119905)
+
31205832
161205832minus 1
coth2120583(119909 minus 1
radicminus161205832+ 1
119905)
1199082(119909 119905) = ∓
3 (119881 + 1) (119881 minus 1)radic120573 + 2
8
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
tanh2120583
times(119909 minus
1
radicminus161205832+ 1
119905)
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
16
coth2120583
times(119909 minus
1
radicminus161205832+ 1
119905)
1199063(119909 119905) =
1205832
41205832+ 1
minus
31205832
41205832+ 1
tanh2120583(119909 minus 1
radic41205832+ 1
119905)
1199083(119909 119905) = ∓
(119881 + 1) (119881 minus 1)radic120573 + 2
4
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
times tanh2120583(119909 minus 1
radic41205832+ 1
119905)
1199064(119909 119905) = minus
31205832
41205832minus 1
+
31205832
41205832minus 1
tanh2120583(119909 minus 1
radicminus41205832+ 1
119905)
1199084(119909 119905) = ∓
3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
plusmn
3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
times tanh2120583(119909 minus 1
radicminus41205832+ 1
119905)
1199065(119909 119905) = minus
31205832
41205832minus 1
+
31205832
41205832minus 1
coth2120583(119909 minus 1
radicminus41205832+ 1
119905)
1199085(119909 119905) = ∓
3 (119881 + 1) (119881 minus 1)radic120573 + 2
4
plusmn 3
(119881 + 1) (119881 minus 1)radic120573 + 2
4
times coth2120583(119909 minus 1
radicminus41205832+ 1
119905)
(40)
35 The Fifth Coupled We finally consider the coupled IBqequations
119906119905119905= 120573119906119909119909minus
120573
2
(1199062)119909119909+
120573
2
(1199082)119909119909+ 1205722119906119909119909119905119905
119908119905119905= 120574(119906119908)
119909119909+ 120588(119908
2)119909119909+ 1205722119908119909119909119905119905
(41)
12 Journal of Applied Mathematics
This system of equations (41) can be converted to the follow-ing system of ODEs by using the wave variable 120585 = 119909 minus 119881119905
(21198812minus 2120573)119880 + 120573119880
2minus 120573119882
2minus 21205722119881211988010158401015840= 0
1198812119882minus 120574119880119882 minus 120588119882
2minus 1205722119881211988210158401015840= 0
(42)
Using the balancing procedure in (42) gives
2119872 = 119872 +119873
2119873 = 119872 +119873
(43)
so that
119872 = 119873 = 2 (44)
We consider solutions in the following form
119880 (120583120585) = 119878 (119884) =
2
sum
119896=0
119886119896119884119896+
2
sum
119896=1
119887119896119884minus119896
119882 (120583120585) = 119878 (119884) =
2
sum
119896=0
119888119896119884119896+
2
sum
119896=1
119889119896119884119896
(45)
Substituting (45) into (42) collecting the coefficients of 119884and solving the resulting system we find the following setsof solutions
1198860= (minus120573 (8120588119888
21205732120574 minus 8120573120574
21205881198882minus 18120588
21205732+ 24120573120588
31198882
+1212057321205742minus 16120574
31205881198882+ 24120588
31198882120574 minus 3120574120573
3))
times (4 (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882+ 312058821205733
+ 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882minus 312057331205742minus 312057431205732))
minus1
1198862=
minus (41205881198882120573 + 4119888
2120588120574 minus 3120573
2)
4120574 (120583 + 120573)
1198880= (120573(120588
212057321198882+ 120574211988821205732+ 12057311988821205743+ 2120574212058821198882
+312057412058821205731198882minus
3
2
1205881205732120574 minus
3
4
1205881205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882+ 312058821205733
+ 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882minus 312057331205742minus 312057431205732)
minus1
1198882= (plusmn3 (minus 120573120588 + (minus120588
21205732+ 41205743120573 minus 4120574
21198814
minus 412057431198812+ 212057321205742
minus41205882120574120573 + 4120588
21198814+ 412058821198812120574)
12
))
times (4 (1205742minus 1205882))
minus1
119881 =
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
120583 =
radicminus120573 (2120574 + 120573)
2120572 (2120574 + 120573)
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860= (120573(6120588119888
21205732120574 + 10120573120574
21205881198882+
3
2
12058821205732+ 2120573120588
31198882
+312057321205742+ 412057431205881198882+ 212058831198882120574 +
9
4
1205741205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882minus 312058821205733
minus 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882+ 312057331205742+ 312057431205732)
minus1
1198862=
minus (41205881198882120573 + 4119888
2120588120574 + 3120573
2)
4120574 (120583 + 120573)
1198880= (120573(minus 3120588
212057321198882minus 3120574211988821205732minus 312057311988821205743
minus6120574212058821198882minus 9120574120588
21205731198882minus
9
2
1205881205732120574 minus
9
4
1205881205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882
minus 312058821205733minus 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882+ 312057331205742+ 312057431205732)
minus1
1198882= (plusmn3 (120573120588 + (minus120588
21205732+ 41205743120573 minus 4120574
21198814
minus 412057431198812+ 212057321205742minus 41205882120574120573
+412058821198814+ 412058821198812120574)
12
))
times (4 (1205742minus 1205882))
minus1
119881 =
radic120573 (412057221205832+ 1)
412057221205832+ 1
120583 =
radic120573 (2120574 + 120573)
2120572 (2120574 + 120573)
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
(46)
Journal of Applied Mathematics 13
In view of this we obtain the following travelling wavesolutions
1199061(119909 119905) = (minus120573 (8120588119888
21205732120574 minus 8120573120574
21205881198882minus 18120588
21205732+ 24120573120588
31198882
+1212057321205742minus 16120574
31205881198882+ 24120588
31198882120574 minus 3120574120573
3))
times (4 (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882+ 312058821205733
+ 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882minus 312057331205742minus 312057431205732))
minus1
minus
(41205881198882120573 + 4119888
2120588120574 minus 3120573
2)
4120574 (120583 + 120573)
tanh2radicminus120573 (2120574 + 120573)
2120572 (2120574 + 120573)
times(119909 minus
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
119905)
1199081(119909 119905) = (120573 (120588
212057321198882+ 120574211988821205732+ 12057311988821205743+ 2120574212058821198882
+312057412058821205731198882minus
3
2
1205881205732120574 minus
3
4
1205881205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882+ 312058821205733
+ 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882minus 312057331205742minus 312057431205732)
minus1
plusmn (3 (minus 120573120588 + (minus12058821205732+ 41205743120573 minus 4120574
21198814
minus 412057431198812+ 212057321205742minus 41205882120574120573
+412058821198814+ 412058821198812120574)
12
))
times (4 (1205742minus 1205882))
minus1
times tanh2radicminus120573 (2120574 + 120573)
2120572 (2120574 + 120573)
times(119909 minus
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
119905)
1199062(119909 119905) = (120573 (6120588119888
21205732120574 + 10120573120574
21205881198882+
3
2
12058821205732+ 2120573120588
31198882
+312057321205742+ 412057431205881198882+ 212058831198882120574 +
9
4
1205741205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882minus 312058821205733
minus 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882+ 312057331205742+ 312057431205732)
minus1
minus
(41205881198882120573 + 4119888
2120588120574 + 3120573
2)
4120574 (120583 + 120573)
tanh2radic120573 (2120574 + 120573)
2120572 (2120574 + 120573)
times(119909 minus
radic120573 (412057221205832+ 1)
412057221205832+ 1
119905)
1199082(119909 119905) = (120573 (minus 3120588
212057321198882minus 3120574211988821205732minus 312057311988821205743
minus6120574212058821198882minus 9120574120588
21205731198882minus
9
2
1205881205732120574 minus
9
4
1205881205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882minus 312058821205733
minus 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882+ 312057331205742+ 312057431205732)
minus1
plusmn (3 (120573120588 + (minus12058821205732+ 41205743120573 minus 4120574
21198814
minus 412057431198812+ 212057321205742minus 41205882120574120573
+412058821198814+ 412058821198812120574)
12
))
times (4 (1205742minus 1205882))
minus1
times tanh2radic120573 (2120574 + 120573)
2120572 (2120574 + 120573)
(119909 minus
radic120573 (412057221205832+ 1)
412057221205832+ 1
119905)
(47)
4 Conclusion
In summary the tanh-coth method has been successfullyimplemented to find new travelling wave solutions for severalkinds of two coupled improved Boussinesq equations Thesolutions that have been found can be seen as an expansionand verification of the previously known data However thisstudy confirmed again the belief that the tanh-coth methodis a powerful and reliable technique to handle nonlineardispersive-dissipative equations and system of equationsThroughout the work Maple has been used to overcome thetedious algebraic calculations
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] A De Godefroy ldquoBlow up of solutions of a generalized Boussi-nesq equationrdquo IMA Journal of AppliedMathematics vol 60 no2 pp 123ndash138 1998
14 Journal of Applied Mathematics
[2] S Wang and M Li ldquoThe Cauchy problem for coupled IMBqequationsrdquo IMA Journal of Applied Mathematics vol 74 no 5pp 726ndash740 2009
[3] N Duruk H A Erbay and A Erkip ldquoBlow-up and globalexistence for a general class of nonlocal nonlinear coupled waveequationsrdquo Journal of Differential Equations vol 250 no 3 pp1448ndash1459 2011
[4] G Chen and H Zhang ldquoInitial boundary value problem for asystem of generalized IMBq equationsrdquoMathematical Methodsin the Applied Sciences vol 27 no 5 pp 497ndash518 2004
[5] P Rosenau ldquoDynamics of dense latticesrdquo Physical Review B vol36 no 11 pp 5868ndash5876 1987
[6] P L Christiansen P S Lomdahl andVMuto ldquoOn aToda latticemodel with a transversal degree of freedomrdquo Nonlinearity vol4 no 2 pp 477ndash501 1991
[7] S K Turitsyn ldquoOn a Toda lattice model with a transversaldegree of freedom Sufficient criterion of blow-up in thecontinuum limitrdquo Physics Letters A vol 173 no 3 pp 267ndash2691993
[8] R L Pego P Smereka and M I Weinstein ldquoOscillatoryinstability of solitary waves in a continuum model of latticevibrationsrdquo Nonlinearity vol 8 no 6 pp 921ndash941 1995
[9] X Fan and L Tian ldquoCompacton solutions and multiple com-pacton solutions for a continuum Toda lattice modelrdquo ChaosSolitons and Fractals vol 29 no 4 pp 882ndash894 2006
[10] A M Wazwaz ldquoThe Hirotarsquos bilinear method and the tanh-coth method for multiple-soliton solutions of the Sawada-Kotera-Kadomtsev-Petviashvili equationrdquoAppliedMathematicsand Computation vol 200 no 1 pp 160ndash166 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Journal of Applied Mathematics
This system of equations (41) can be converted to the follow-ing system of ODEs by using the wave variable 120585 = 119909 minus 119881119905
(21198812minus 2120573)119880 + 120573119880
2minus 120573119882
2minus 21205722119881211988010158401015840= 0
1198812119882minus 120574119880119882 minus 120588119882
2minus 1205722119881211988210158401015840= 0
(42)
Using the balancing procedure in (42) gives
2119872 = 119872 +119873
2119873 = 119872 +119873
(43)
so that
119872 = 119873 = 2 (44)
We consider solutions in the following form
119880 (120583120585) = 119878 (119884) =
2
sum
119896=0
119886119896119884119896+
2
sum
119896=1
119887119896119884minus119896
119882 (120583120585) = 119878 (119884) =
2
sum
119896=0
119888119896119884119896+
2
sum
119896=1
119889119896119884119896
(45)
Substituting (45) into (42) collecting the coefficients of 119884and solving the resulting system we find the following setsof solutions
1198860= (minus120573 (8120588119888
21205732120574 minus 8120573120574
21205881198882minus 18120588
21205732+ 24120573120588
31198882
+1212057321205742minus 16120574
31205881198882+ 24120588
31198882120574 minus 3120574120573
3))
times (4 (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882+ 312058821205733
+ 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882minus 312057331205742minus 312057431205732))
minus1
1198862=
minus (41205881198882120573 + 4119888
2120588120574 minus 3120573
2)
4120574 (120583 + 120573)
1198880= (120573(120588
212057321198882+ 120574211988821205732+ 12057311988821205743+ 2120574212058821198882
+312057412058821205731198882minus
3
2
1205881205732120574 minus
3
4
1205881205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882+ 312058821205733
+ 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882minus 312057331205742minus 312057431205732)
minus1
1198882= (plusmn3 (minus 120573120588 + (minus120588
21205732+ 41205743120573 minus 4120574
21198814
minus 412057431198812+ 212057321205742
minus41205882120574120573 + 4120588
21198814+ 412058821198812120574)
12
))
times (4 (1205742minus 1205882))
minus1
119881 =
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
120583 =
radicminus120573 (2120574 + 120573)
2120572 (2120574 + 120573)
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
1198860= (120573(6120588119888
21205732120574 + 10120573120574
21205881198882+
3
2
12058821205732+ 2120573120588
31198882
+312057321205742+ 412057431205881198882+ 212058831198882120574 +
9
4
1205741205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882minus 312058821205733
minus 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882+ 312057331205742+ 312057431205732)
minus1
1198862=
minus (41205881198882120573 + 4119888
2120588120574 + 3120573
2)
4120574 (120583 + 120573)
1198880= (120573(minus 3120588
212057321198882minus 3120574211988821205732minus 312057311988821205743
minus6120574212058821198882minus 9120574120588
21205731198882minus
9
2
1205881205732120574 minus
9
4
1205881205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882
minus 312058821205733minus 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882+ 312057331205742+ 312057431205732)
minus1
1198882= (plusmn3 (120573120588 + (minus120588
21205732+ 41205743120573 minus 4120574
21198814
minus 412057431198812+ 212057321205742minus 41205882120574120573
+412058821198814+ 412058821198812120574)
12
))
times (4 (1205742minus 1205882))
minus1
119881 =
radic120573 (412057221205832+ 1)
412057221205832+ 1
120583 =
radic120573 (2120574 + 120573)
2120572 (2120574 + 120573)
1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0
(46)
Journal of Applied Mathematics 13
In view of this we obtain the following travelling wavesolutions
1199061(119909 119905) = (minus120573 (8120588119888
21205732120574 minus 8120573120574
21205881198882minus 18120588
21205732+ 24120573120588
31198882
+1212057321205742minus 16120574
31205881198882+ 24120588
31198882120574 minus 3120574120573
3))
times (4 (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882+ 312058821205733
+ 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882minus 312057331205742minus 312057431205732))
minus1
minus
(41205881198882120573 + 4119888
2120588120574 minus 3120573
2)
4120574 (120583 + 120573)
tanh2radicminus120573 (2120574 + 120573)
2120572 (2120574 + 120573)
times(119909 minus
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
119905)
1199081(119909 119905) = (120573 (120588
212057321198882+ 120574211988821205732+ 12057311988821205743+ 2120574212058821198882
+312057412058821205731198882minus
3
2
1205881205732120574 minus
3
4
1205881205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882+ 312058821205733
+ 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882minus 312057331205742minus 312057431205732)
minus1
plusmn (3 (minus 120573120588 + (minus12058821205732+ 41205743120573 minus 4120574
21198814
minus 412057431198812+ 212057321205742minus 41205882120574120573
+412058821198814+ 412058821198812120574)
12
))
times (4 (1205742minus 1205882))
minus1
times tanh2radicminus120573 (2120574 + 120573)
2120572 (2120574 + 120573)
times(119909 minus
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
119905)
1199062(119909 119905) = (120573 (6120588119888
21205732120574 + 10120573120574
21205881198882+
3
2
12058821205732+ 2120573120588
31198882
+312057321205742+ 412057431205881198882+ 212058831198882120574 +
9
4
1205741205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882minus 312058821205733
minus 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882+ 312057331205742+ 312057431205732)
minus1
minus
(41205881198882120573 + 4119888
2120588120574 + 3120573
2)
4120574 (120583 + 120573)
tanh2radic120573 (2120574 + 120573)
2120572 (2120574 + 120573)
times(119909 minus
radic120573 (412057221205832+ 1)
412057221205832+ 1
119905)
1199082(119909 119905) = (120573 (minus 3120588
212057321198882minus 3120574211988821205732minus 312057311988821205743
minus6120574212058821198882minus 9120574120588
21205731198882minus
9
2
1205881205732120574 minus
9
4
1205881205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882minus 312058821205733
minus 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882+ 312057331205742+ 312057431205732)
minus1
plusmn (3 (120573120588 + (minus12058821205732+ 41205743120573 minus 4120574
21198814
minus 412057431198812+ 212057321205742minus 41205882120574120573
+412058821198814+ 412058821198812120574)
12
))
times (4 (1205742minus 1205882))
minus1
times tanh2radic120573 (2120574 + 120573)
2120572 (2120574 + 120573)
(119909 minus
radic120573 (412057221205832+ 1)
412057221205832+ 1
119905)
(47)
4 Conclusion
In summary the tanh-coth method has been successfullyimplemented to find new travelling wave solutions for severalkinds of two coupled improved Boussinesq equations Thesolutions that have been found can be seen as an expansionand verification of the previously known data However thisstudy confirmed again the belief that the tanh-coth methodis a powerful and reliable technique to handle nonlineardispersive-dissipative equations and system of equationsThroughout the work Maple has been used to overcome thetedious algebraic calculations
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] A De Godefroy ldquoBlow up of solutions of a generalized Boussi-nesq equationrdquo IMA Journal of AppliedMathematics vol 60 no2 pp 123ndash138 1998
14 Journal of Applied Mathematics
[2] S Wang and M Li ldquoThe Cauchy problem for coupled IMBqequationsrdquo IMA Journal of Applied Mathematics vol 74 no 5pp 726ndash740 2009
[3] N Duruk H A Erbay and A Erkip ldquoBlow-up and globalexistence for a general class of nonlocal nonlinear coupled waveequationsrdquo Journal of Differential Equations vol 250 no 3 pp1448ndash1459 2011
[4] G Chen and H Zhang ldquoInitial boundary value problem for asystem of generalized IMBq equationsrdquoMathematical Methodsin the Applied Sciences vol 27 no 5 pp 497ndash518 2004
[5] P Rosenau ldquoDynamics of dense latticesrdquo Physical Review B vol36 no 11 pp 5868ndash5876 1987
[6] P L Christiansen P S Lomdahl andVMuto ldquoOn aToda latticemodel with a transversal degree of freedomrdquo Nonlinearity vol4 no 2 pp 477ndash501 1991
[7] S K Turitsyn ldquoOn a Toda lattice model with a transversaldegree of freedom Sufficient criterion of blow-up in thecontinuum limitrdquo Physics Letters A vol 173 no 3 pp 267ndash2691993
[8] R L Pego P Smereka and M I Weinstein ldquoOscillatoryinstability of solitary waves in a continuum model of latticevibrationsrdquo Nonlinearity vol 8 no 6 pp 921ndash941 1995
[9] X Fan and L Tian ldquoCompacton solutions and multiple com-pacton solutions for a continuum Toda lattice modelrdquo ChaosSolitons and Fractals vol 29 no 4 pp 882ndash894 2006
[10] A M Wazwaz ldquoThe Hirotarsquos bilinear method and the tanh-coth method for multiple-soliton solutions of the Sawada-Kotera-Kadomtsev-Petviashvili equationrdquoAppliedMathematicsand Computation vol 200 no 1 pp 160ndash166 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 13
In view of this we obtain the following travelling wavesolutions
1199061(119909 119905) = (minus120573 (8120588119888
21205732120574 minus 8120573120574
21205881198882minus 18120588
21205732+ 24120573120588
31198882
+1212057321205742minus 16120574
31205881198882+ 24120588
31198882120574 minus 3120574120573
3))
times (4 (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882+ 312058821205733
+ 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882minus 312057331205742minus 312057431205732))
minus1
minus
(41205881198882120573 + 4119888
2120588120574 minus 3120573
2)
4120574 (120583 + 120573)
tanh2radicminus120573 (2120574 + 120573)
2120572 (2120574 + 120573)
times(119909 minus
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
119905)
1199081(119909 119905) = (120573 (120588
212057321198882+ 120574211988821205732+ 12057311988821205743+ 2120574212058821198882
+312057412058821205731198882minus
3
2
1205881205732120574 minus
3
4
1205881205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882+ 312058821205733
+ 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882minus 312057331205742minus 312057431205732)
minus1
plusmn (3 (minus 120573120588 + (minus12058821205732+ 41205743120573 minus 4120574
21198814
minus 412057431198812+ 212057321205742minus 41205882120574120573
+412058821198814+ 412058821198812120574)
12
))
times (4 (1205742minus 1205882))
minus1
times tanh2radicminus120573 (2120574 + 120573)
2120572 (2120574 + 120573)
times(119909 minus
radicminus120573 (412057221205832minus 1)
412057221205832minus 1
119905)
1199062(119909 119905) = (120573 (6120588119888
21205732120574 + 10120573120574
21205881198882+
3
2
12058821205732+ 2120573120588
31198882
+312057321205742+ 412057431205881198882+ 212058831198882120574 +
9
4
1205741205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882minus 312058821205733
minus 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882+ 312057331205742+ 312057431205732)
minus1
minus
(41205881198882120573 + 4119888
2120588120574 + 3120573
2)
4120574 (120583 + 120573)
tanh2radic120573 (2120574 + 120573)
2120572 (2120574 + 120573)
times(119909 minus
radic120573 (412057221205832+ 1)
412057221205832+ 1
119905)
1199082(119909 119905) = (120573 (minus 3120588
212057321198882minus 3120574211988821205732minus 312057311988821205743
minus6120574212058821198882minus 9120574120588
21205731198882minus
9
2
1205881205732120574 minus
9
4
1205881205733))
times (minus812057312058831198882120574 minus 4120588
312057421198882minus 4120573212058831198882minus 312058821205733
minus 312058821205741205732+ 4120588120574
41198882+ 4120573212057421205881198882
+812057312057431205881198882+ 312057331205742+ 312057431205732)
minus1
plusmn (3 (120573120588 + (minus12058821205732+ 41205743120573 minus 4120574
21198814
minus 412057431198812+ 212057321205742minus 41205882120574120573
+412058821198814+ 412058821198812120574)
12
))
times (4 (1205742minus 1205882))
minus1
times tanh2radic120573 (2120574 + 120573)
2120572 (2120574 + 120573)
(119909 minus
radic120573 (412057221205832+ 1)
412057221205832+ 1
119905)
(47)
4 Conclusion
In summary the tanh-coth method has been successfullyimplemented to find new travelling wave solutions for severalkinds of two coupled improved Boussinesq equations Thesolutions that have been found can be seen as an expansionand verification of the previously known data However thisstudy confirmed again the belief that the tanh-coth methodis a powerful and reliable technique to handle nonlineardispersive-dissipative equations and system of equationsThroughout the work Maple has been used to overcome thetedious algebraic calculations
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] A De Godefroy ldquoBlow up of solutions of a generalized Boussi-nesq equationrdquo IMA Journal of AppliedMathematics vol 60 no2 pp 123ndash138 1998
14 Journal of Applied Mathematics
[2] S Wang and M Li ldquoThe Cauchy problem for coupled IMBqequationsrdquo IMA Journal of Applied Mathematics vol 74 no 5pp 726ndash740 2009
[3] N Duruk H A Erbay and A Erkip ldquoBlow-up and globalexistence for a general class of nonlocal nonlinear coupled waveequationsrdquo Journal of Differential Equations vol 250 no 3 pp1448ndash1459 2011
[4] G Chen and H Zhang ldquoInitial boundary value problem for asystem of generalized IMBq equationsrdquoMathematical Methodsin the Applied Sciences vol 27 no 5 pp 497ndash518 2004
[5] P Rosenau ldquoDynamics of dense latticesrdquo Physical Review B vol36 no 11 pp 5868ndash5876 1987
[6] P L Christiansen P S Lomdahl andVMuto ldquoOn aToda latticemodel with a transversal degree of freedomrdquo Nonlinearity vol4 no 2 pp 477ndash501 1991
[7] S K Turitsyn ldquoOn a Toda lattice model with a transversaldegree of freedom Sufficient criterion of blow-up in thecontinuum limitrdquo Physics Letters A vol 173 no 3 pp 267ndash2691993
[8] R L Pego P Smereka and M I Weinstein ldquoOscillatoryinstability of solitary waves in a continuum model of latticevibrationsrdquo Nonlinearity vol 8 no 6 pp 921ndash941 1995
[9] X Fan and L Tian ldquoCompacton solutions and multiple com-pacton solutions for a continuum Toda lattice modelrdquo ChaosSolitons and Fractals vol 29 no 4 pp 882ndash894 2006
[10] A M Wazwaz ldquoThe Hirotarsquos bilinear method and the tanh-coth method for multiple-soliton solutions of the Sawada-Kotera-Kadomtsev-Petviashvili equationrdquoAppliedMathematicsand Computation vol 200 no 1 pp 160ndash166 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
14 Journal of Applied Mathematics
[2] S Wang and M Li ldquoThe Cauchy problem for coupled IMBqequationsrdquo IMA Journal of Applied Mathematics vol 74 no 5pp 726ndash740 2009
[3] N Duruk H A Erbay and A Erkip ldquoBlow-up and globalexistence for a general class of nonlocal nonlinear coupled waveequationsrdquo Journal of Differential Equations vol 250 no 3 pp1448ndash1459 2011
[4] G Chen and H Zhang ldquoInitial boundary value problem for asystem of generalized IMBq equationsrdquoMathematical Methodsin the Applied Sciences vol 27 no 5 pp 497ndash518 2004
[5] P Rosenau ldquoDynamics of dense latticesrdquo Physical Review B vol36 no 11 pp 5868ndash5876 1987
[6] P L Christiansen P S Lomdahl andVMuto ldquoOn aToda latticemodel with a transversal degree of freedomrdquo Nonlinearity vol4 no 2 pp 477ndash501 1991
[7] S K Turitsyn ldquoOn a Toda lattice model with a transversaldegree of freedom Sufficient criterion of blow-up in thecontinuum limitrdquo Physics Letters A vol 173 no 3 pp 267ndash2691993
[8] R L Pego P Smereka and M I Weinstein ldquoOscillatoryinstability of solitary waves in a continuum model of latticevibrationsrdquo Nonlinearity vol 8 no 6 pp 921ndash941 1995
[9] X Fan and L Tian ldquoCompacton solutions and multiple com-pacton solutions for a continuum Toda lattice modelrdquo ChaosSolitons and Fractals vol 29 no 4 pp 882ndash894 2006
[10] A M Wazwaz ldquoThe Hirotarsquos bilinear method and the tanh-coth method for multiple-soliton solutions of the Sawada-Kotera-Kadomtsev-Petviashvili equationrdquoAppliedMathematicsand Computation vol 200 no 1 pp 160ndash166 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of