Boussinesq Solitons

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SOOCHOW JOURNAL OF MATHEMATICS

Volume 30, No. 1, pp. 91-101, January 2004

TRAVELLING AND PERIODIC WAVE SOLUTIONS

OF A CLASSICAL BOUSSINESQ SYSTEM

BY

M. INC

Abstract. In this paper, we use of the modified extended tanh-function and theJacobi elliptic function methods to obtain travelling wave and Jacobi doubly peri-odic wave solutions for the classical Boussinesq system. In addition, the propertiesof this equation shown with figures.

1. Introduction

The investigation of the travelling wave solutions play an important role

in nonlinear science. These solutions may well describe various phenomena in

nature, such as vibrations, solitons and propagation with a finite speed. Thewave phenomena observed in fluid dynamics, plasma and elastic media. Vari-

ous methods for obtaining explicit travelling solitary wave solutions to nonlinear

evolution equations have been proposed. In recent years, directly searching for

exact solutions of nonlinear PDEs has become more and more attractive partly

due to the availability of computer symbolic systems like Maple or Mathematica

which allow us to perform some complicated and tedious algebraic calculation

on a computer, as well as help us to find new exact solutions of PDEs ([1-8]).

One of most effectively straightforward methods to construct exact solution of

PDE is extended tanh-function method ([9-13]). Recently, Elwakil et al. [14, 15]developed a modified extended tanh-function method for solving nonlinear PDEs.

Let us simply describe the modified extended tanh-function method.

Received April 1, 2003; revised September 1, 2003.AMS Subject Classification. 35C05, 35Q51, 35B10.Key words. modified extended tanh-function method, Jacobi elliptic function method, soli-

tary wave solution, nonlinear equation, periodic wave solution.

91


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For given a nonlinear equation

H(u, ut, ux, uxx, uxt, . . .) = 0, (1)

when we look for its travelling wave solutions, the first step is to introduce the

wave transformation u = U() , = x + t and change (1) to an ordinary differ-

ential equation

H

U, U

, U

, . . .

= 0. (2)

The next crucial step is to introduce a new variable = () which is a solutionof the Riccati equation

= b + 2, (3)

where b is a parameter to be determined, = () ,

= d/d. Then we

propose the following series expansion as a solution of (1):

u (x, t) = U() =mi=0

aii +

mi=0

bii, (4)

where the positive integer m can be determined by balancing the highest deriva-

tive term with nonlinear terms in (2). Substituting (3) and (4) into (2) will get a

system of algebraic equations with respect to ai, bi, b and (where i = 0, 1, . . . , m)

because all the coefficients ofi have to vanish. With the aid of Mathematica, one

can determine ai, bi, b and . The Riccati equation (3) has the general solutions

=

b tanh

bb coth

b for b < 0, (5)

=

1

for b = 0, (6)

=

b tan

b

b cot

b for b > 0. (7)

We now describe the Jacobi elliptic function method.

We again consider Eq.(2). The fact that the solutions of many nonlinear

equations can be expressed as a finite series of Jacobi elliptic sine, cosine and the


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TRAVELLING AND PERIODIC WAVE SOLUTIONS 93

third kind of Jacobi elliptic functions expansions can be written as, respectively

[16-18]:

u () =n

j=0

ajsnj,

u () =n

j=0

bjcnj,

u () =n

j=0

cjdnj.

(8)

Notice that the highest power order of U() is equal to n,

O (U()) = n, (9)

and the highest power order of dU/d can be taken as

O

dU

d

= n + 1. (10)

We have

O

dpU

dp

= n + p, p = 1, 2, 3, . . . , (11)

and

O

Uq

dpU

dp

= (q+ 1) n + p, q = 0, 1, 2, . . . , (12)

so n can be obtained by balancing the derivative term of the highest order with

the nonlinear term in Eq.(2). c, a0, . . . , an; b0, b1, . . . , bn are parameters to be

determined. Substituting (8) into (2) will yield a set of algebraic equations for

c, a0, . . . , an; b0, b1, . . . , bn because all coefficients of snj and cnj have to vanish.

From these relations, c, a0, . . . , an; b0, b1, . . . , bn can be obtained. Therefore, the

travelling solitary wave solutions are obtained.

It is known that there are the following relations between elliptic functions:

cn2= 1 sin2 , sn2+ cn2 = 1,dn2= 1 m2sn2 , d

dsn = cndn, (13)

d

dcn=sndn , d

ddn = m2sncn,

where m is the modulus 0 < m < 1.


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When m 1, the Jacobi functions degenerate to the hyperbolic functions,i.e.,

sn tanh , cn sec h , dn sec h. (14)When m 0, the Jacobi functions degenerate to the triangular functions, i.e.,

sn sin , cn cos , dn 1. (15)

In [19], three sets of model equations are derived for modelling nonlinear and

dispersive long gravity waves travelling in two horizontal directions on shallow

waters of uniform depth. Omitting the higher order terms, one of these equations,

the Wu-Zhang (WZ) equation, can be written as

ut + uux + vuy + wx = 0,

vt + uvx + vvy + wy = 0,

wt + (uw)x + (uw)y +1

3(u3x + uxyy + vxxy + v3y) = 0,

(16)

where w 1 is the elevation of the water wave, u is the surface velocity of wateralong the x direction, and v is the surface velocity of water along the y direction.

By scaling transformation and symmetry reduction, Eq.(16) can be reduced to

the (1+1)-dimensional dispersive long wave equation ([19-23])vt + vvy + wy = 0,

wt + (wu)y +1

3vyyy = 0.

(17)

A good understanding of all solutions of Eq.(16) is very helpful for coastal and

civil engineers to apply the nonlinear water wave model in a harbor and coastal

design. In [20], some special type soliton solutions for Eq.(16) is derived directly

by using the standard and nonstandard truncation of the WTCs approach and

the modified Contes invariant Painleve expansion for the WZ equation. In [21],

Zheng et al. obtain known solitary wave solutions, other new and more generalsolutions of Eq.(17) by using the generalized extended tanh-function method with

a new ansatze. Zhang and Li [22] study bidirectional solitons on water of Eq.(17)

by using the Darboux transformation method. In this paper, we consider the

following classical Boussinesq systemt + [(1 + ) u]x +

1

4uxxx = 0,

ut + uux + x = 0,(18)


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where is the elevation of water wave, and u is the surface velocity of water along

the xdirection ([24-26]). Li et al. [24] gave two basic Darboux transformationsof a spectral problem associated with the Broer-Kaup system and used them to

generate new solutions of the Eq.(18). Recently, Li and Zhang [25] presented

the third kind of Darboux transformation of Eq.(18) and they discussed its rela-

tionship with the two basic Darboux transformations. Thus, they obtained the

solutions of multiple soliton interactions. More recently, Zhang et al. [26] maked

a simple ansatz to the solutions for Eq.(18) and they obtained the general ex-

plicit solutions. Here we use the modified extended tanh-function and the Jacobielliptic function methods for obtaining new travelling wave and Jacobi doubly

periodic wave solutions of Eq.(18).

2. Travelling and Periodic Wave Solutions of Eq.(18)

To seek the travelling wave and Jacobi doubly periodic wave solutions of

Eq.(18). We make the travelling wave transformation (x, t) = (x) , u (x, t) =

U() , = x t and we change Eq.(18) into the form

() + U() () + U() + 14

U

() = 0, (19)

() U() + 12

U2 () = 0, (20)

where the prime denotes d/d ([21]). Inserting Eq.(20) into Eq.(19) leads to an

ordinary differential equation

3

2U2 () 1

2U3 () 2U() + U() + 1

4U

() = 0. (21)

Balancing U

with U3 yields m = 1. Therefore, we have

U = a0 + a1 + b0 + b11. (22)

Substituting Eq.(22) into Eq.(21) and making use of Eq.(3), with the help of

Mathematica we get a system of algebraic equations for a0, a1, b0, b1, b and :

3

2a20 + 3a0b0 + 3a1b1 +

3

2b20

3

2a20b0

1

2a30 3a1b1a0 3a0a1b1

12

b30 3

2a0b

20 2a0 2b0 + a0 + b0 = 0,


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3a0a1 + 3a1b0 32

a1b20

3

2a21b13a0a1b0

3

2a20a1 + a1

12

+

1

2a1b = 0,

3a0b1 + 3b0b1 3

2b1a

20

1

2b31

3

2a1b

20

3

2b20b1

3

2a1b

21 3a0b0b1

2b1 + b1 +1

2bb1 = 0,

a21 a21b0 a0a21 = 0, b21 b21b0 a0b21 = 0,1

2a31 +

1

2a1 = 0,

1

2b2b1 = 0.

From the out put of symbolic computation software Mathematica, we obtain

b = 0, a1 = i, a0 = b0 and b1 = 13

i

2 + 1

, (23)

b = 0, b0 = b1 = 0, a1 = i, a0 =1

3

2 1

, b =

1

3

44 162 + 6 + 1

,

(24)

b = 0, b0 = b1 = 0, a1 = i, a0 = and b = 2 2, (25)where b0 and b are arbitrary constants. Since b is a arbitrary parameter, according

to (5)-(7) and (23)-(25), we obtain three kinds of travelling wave solutions for the

Eq.(18): Soliton solutions with b < 0,

u1 =1

3

2 1

ib tanh

b (x t)

, (26)

u2 =1

3

2 1

ib coth

b (x t)

, (27)

where b = 13

44 162 + 6 + 1 .

u3 = i

2 + 2 tanh

2 + 2 (x t)

, (28)

u4 = i

2 + 2 coth

2 + 2 (x t)

. (29)

Periodic solutions with b > 0,

u5 =1

3

2 1

i

b tan

b (x t)

, (30)

u6 =1

3

2 1

i

b cot

b (x t)

, (31)


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where b = 13

44 162 + 6 + 1 .

u7 =

2 + 2 tan

2 + 2i (x t)

, (32)

u8 =

2 + 2 cot

2 + 2i (x t)

. (33)

A rational solution with b = 0,

u9 = b0 i

2 + 4

3 (x t) . (34)

According to the Jacobi elliptic function method, we get the following Jacobidoubly periodic wave solutions for Eq.(18):

u10 =

1

2

1 + m2

2

1/2msn

x

1

2

1 + m2

2

1/2t

, (35)

u11 =

1

2

1 2m2

2

1/2mcn

x

1

2

1 2m2

2

1/2t

, (36)

u12 =

1

2

2m2

2

1/2 dn

x

1

2

2m2

2

1/2t

, (37)

u13 =

1

2

2m2

2

1/2

cs

x

1

2

2m2

2

1/2

t

. (38)

And (35)-(38) corresponding travelling wave solutions are

u14 = i + tanh (x it) , (39)

u15 =

3i + sec h

x

3it

, (40)

u16 = i

5

2 csc h

x

3

2it

. (41)

Remark. It is easily seen that u1, . . . , u8 are like the solutions of Zheng etal. [21] but these solutions are not the same. We knowledge, the obtained solu-

tions of Eq.(18), u9, u14, u15 and u16 were not found by the modified extended

tanh-function method ([14, 15]) and the generalized extended tanh-function

method ([21]). In addition, we obtain some new complex formal solutions and

Jacobi doubly periodic wave solutions in the paper. To compare the new for-

mal solutions for Eq.(18) with the known formal solutions, we draw some plots


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for some formal solutions of WZ equation (18). The properties of some formal

solutions are shown in Figure 1.

Figure 1. The soliton and periodic wave solutions of Eq.(18), where = 1, 1.5.


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Figure 2. The Jacobi doubly periodic wave solutions of Eq.(18).

3. Conclusions

In this paper, the modified extended tanh-function mathod and the Jacobi

elliptic function expansion method are applied to the classical Boussinesq system.

The aim to obtain travelling wave and Jacobi doubly periodic wave solutions of

this equation by using these methods have been achieved. In the fact, the present

methods are readily applicable to a large variety of such nonlinear equations. The

properties of the Jacobi doubly periodic wave solutions are shown in Figure 2.

Our present methods are very easy applied to both differential equations and

linear or nonlinear differential systems.


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Department of Mathematics, Firat University, Elazig 23119 / TURKIYE.

E-mail: [email protected]

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