PRAMANA c© Indian Academy of Sciences Vol. 77, No. 3— journal of September 2011

physics pp. 447–460

Higher-order symmetries and conservation lawsof multi-dimensional Gordon-type equations

S JAMAL1,∗ and A H KARA2

1School of Mathematics, University of the Witwatersrand, Johannesburg, Private Bag 3, Wits 2050,South Africa2School of Mathematics and Centre for Differential Equations, Continuum Mechanics andApplications, University of the Witwatersrand, Johannesburg, Private Bag 3, Wits 2050,South Africa∗Corresponding author. E-mail: sameerahj@hotmail.com

Abstract. In this paper a class of multi-dimensional Gordon-type equations are analysed using amultiplier and homotopy approach to construct conservation laws. The main focus is the analysis ofthe classical versions of the Gordon-type equations and obtaining higher-order variational symmetriesand corresponding conserved quantities. The results are extended to the multi-dimensional Gordon-type equations with the two-dimensional Klein–Gordon equation in particular yielding interestingresults.

Keywords. Conservation laws; multipliers; multi-dimensional Gordon-type equations.

PACS Nos 02.30.Hq; 02.30.Jr; 02.30.Xx; 02.40.Ky

1. Introduction

The higher-order multipliers and conservation laws of the canonical form of the Gordon-type equations,

u XT − k(u) = 0, (1)

can be recast in the well-known classical form,

utt − uxx − k(u) = 0. (2)

The purpose for doing this is to gain an insight into extending the results to the multi-dimensional Gordon-type equations which are somewhat cumbersome if one was to pursuea canonical form. That is, the multi-dimensional Gordon-type equations are best consid-ered as

utt − �u − k(u) = 0, (3)

DOI: 10.1007/s12043-011-0165-5; ePublication: 26 August 2011 447

S Jamal and A H Kara

where � denotes the Laplacian. This equation is of the classical form rather than thecanonical form.

Equation (1) has been extensively studied in terms of their symmetries and variationalproperties [1]. In particular, the sine-Gordon equation u XT − sin u = 0 has been shown tohave higher-order variational symmetries, X = Q∂u . For example,

X1 =(

u X X X + 1

2u3

X

)∂u,

X2 =(

u X X X X X + 5

2u2

X u X X X + 5

2u X u2

X X + 3

8u5

X

)∂u,

X3 =(

uT T T + 1

2u3

T

)∂u . (4)

Of the variational symmetries in (4), the first two lead to the corresponding higher-orderconserved densities,

�T1 = −1

2u2

X X + 1

8u4

X , �T2 = 1

2u2

X X X − 5

4u2

X u2X X + 1

16u6

X . (5)

In a similar way, we investigate the existence of higher-order variational symmetries forother classes of Gordon-type equations with dimension one and two, and we find possibleconserved densities via the multiplier method outlined below. We conclude with someinteresting and unexpected results.

We apply the multiplier approach that leads to a large class of interesting and higher-order conserved flows that would not have been obtained by variational techniques such asNoether’s theorem. In particular, we obtain higher-order multipliers.

We present some of the definitions and notations below. Intrinsic to a Lie algebraictreatment of differential equations is the universal space A (see [1]). The space A isthe vector space of all differential functions of all finite orders and forms an algebra.Consider an r th-order system of partial differential equations of n independent variablesx = (x1, x2, . . . , xn) and m dependent variables u = (u1, u2, . . . , um),

Gμ(x, u, u(1), . . . , u(r)) = 0, μ = 1, . . . , m̃, (6)

where u(1), u(2), . . . , u(r) denote the collections of all first-, second-, . . ., r th-order par-tial derivatives, that is, uα

i = Di (uα), uαi j = D j Di (uα), . . . respectively, with the total

differentiation operator with respect to xi given by

Di = ∂

∂xi+ uα

i

∂

∂uα+ uα

i j

∂

∂uαj

+ · · · , i = 1, . . . , n, (7)

where the summation convention is used whenever appropriate. A current � = (�1, . . . , �n)

is conserved if it satisfies

Di�i = 0 (8)

448 Pramana – J. Phys., Vol. 77, No. 3, September 2011

Higher-order symmetries and conservation laws of Gordon-type equations

along the solutions of (6). It can be shown that every admitted conservation law arises frommultipliers Qμ(x, u, u(1), . . .) such that

QμGμ = Di�i (9)

holds identically (that is, off the solution space) for some current �. The conserved vectormay then be obtained by the homotopy operator (see [2–4]). Other works on symmetriesand conservation laws can be found in [5–8].

Definition 1. The Euler operator for each dependent variable uα is defined by

δ

δuα= ∂

∂uα+

∑s≥1

(−1)s Di1 · · · Dis

∂

∂uαi1···is

, α = 1, . . . , m. (10)

In most literature, a variational problem consists of finding the extrema (maxima orminima) of a functional,

L[u] =∫

�

L(x, u(n))dx,

in some class of functions u = f (x) defined over � where � ⊂ X is an open con-nected subset with smooth boundary ∂� (we consider the Euclidean space with X = Rn).The integrand L(x, u(n)), called the Lagrangian of the variational problem L, is a smoothfunction of x, u and various derivatives of u [1].

Definition 2. The Lie–Bäcklund or generalized operator is given by

X = ξ i ∂

∂xi+ ηα ∂

∂uα, ξ i , ηα ∈ A. (11)

This operator is an abbreviated form of the following infinite formal sum:

X = ξ i ∂

∂xi+ ηα ∂

∂uα+

∑s≥1

ζ αi1...is

∂

∂uαi1...is

, (12)

where the additional coefficients are determined uniquely by the prolongation formulae,

ζ αi = Di (W α) + ξ j uα

i j ,

ζ αi1...is

= Di1 . . . Dis (W α) + ξ j uαj i1...is

, s > 1. (13)

Definition 3. A Lie–Bäcklund operator X of the form (11) is called a variational symme-try if it leaves invariant the functional L[u] = ∫

L(x, u(n))dx . Variational partial differ-ential equations are partial differential equations that admit Lagrangians.

The following theorem defines the condition under which a symmetry is variational.

Theorem 1. For variational partial differential equations, E = 0, where E ∈ A, anevolutionary vector field X = Q∂u is a variational symmetry if and only if X E+ AFQ E =0, where AFQ is the adjoint Fréchet derivative on Q [1].

Pramana – J. Phys., Vol. 77, No. 3, September 2011 449

S Jamal and A H Kara

2. Applications

We now construct symmetries and conservation laws for the classes of eq. (2) discussedabove with particular emphasis on the main eq. (3). Most of the tedious calculations havebeen omitted.

2.1 (1+1) Gordon-type equations

We now consider some special cases of eq. (2) in classical form. In each case, we list oneor two of the higher-order multipliers and conserved densities which arise as a consequenceof the transformation to the classical form. Other multipliers and conserved densities existand may be found in [9].

2.1.1 The Gordon-type equation: uxx −utt − sin u = 0. In converting the equation uxx−utt − sin(u) = 0 into u XT − sin u, we use the transformations,

X = 1

2(x − t), T = 1

2(x + t).

In the reverse direction we use the transformations,

x = X + T, t = T − X,

to obtain higher-order symmetries for the classical equation which we call X̄ . We shallnow illustrate the method using X1 = (u X X X + 1

2 u3X )∂u listed in (4). Since

u X = ux xX + ut tX = ux − ut ,

u X X = uxxxX + uxttX − utxxX − utttX = uxx − 2uxt + utt,

u X X X = uxxxxX + uxxttX − 2(uxtxxX + uxtttX ) + uttxxX + uttttX

= uxxx − 3uxxt + 3uxtt − uttt,

the equivalent of X1 is

X̄1 =(

uxxx − 3uxxt + 3uxtt − uttt + 1

2(ux − ut )

3

)∂u,

with the multiplier,

Q1 = 1

2(ux − ut )

3 + uxxx − uttt − 3utxx + 3uttx,

leading to a conserved vector,

�t1 = 1

8

(u4

t + 4u2tt − 3u3

t ux + 4utttux − 4 cos(u) u2x + 8utt (sin(u) − uxt)

− 16 sin(u) uxt + 3uu2xuxt − 4uxuxtt + 4uuxttt

+ u2t

(−4 cos(u) + 3u2x + 3u (uxt − uxx)

) + 8 sin(u) uxx

− 3uu2xuxx + 8uxtuxx − 4u2

xx − 4uxuxxt − 12uuxxtt + 4uxuxxx

+ ut(−u3

x + ux (8 cos(u) − 6u(uxt − uxx))

− 8 (uxtt − 2uxxt + uxxx))

+ 12uuxxxt − 4uuxxxx),

450 Pramana – J. Phys., Vol. 77, No. 3, September 2011

Higher-order symmetries and conservation laws of Gordon-type equations

�x1 = 1

8

(−4u2tt − 4uutttt − u3

t ux − 8utttux + 4 cos uu2x + u4

x

+ u2t

(4 cos u + 3u2

x − 3u (utt − uxt)) + 16 sin uuxt + 3uu2

x uxt

+ utt(−8 sin u − 3uu2

x + 8uxt) + 16ux uxtt + 12uuxttt − 8 sin uuxx

− 8uxtuxx + 4u2xx − 8ux uxxt − 12uuxxtt

+ ut(4uttt − 3u3

x + ux (−8 cos u + 6u (utt − uxt))

− 4 (uxtt + uxxt − uxxx)) + 4uuxxxt).

2.1.2 The Gordon-type equation: uxx − utt − u = 0. The multiplier,

Q2A = −uxxx + uttt + 3utxx − 3uttx, (14)

leads to the conserved vector,

�t2A = 1

2

(u2

t − u2tt − utttux + u2

x + 2uttuxt + ux uxtt − 2uxtuxx + u2xx

+ ux uxxt − 2ut(ux − uxtt + 2uxxt − uxxx

) − ux uxxx

+ u( − 2utt + 4uxt − uxttt − 2uxx + 3uxxtt − 3uxxxt + uxxxx

)),

�x2A = 1

2

( − u2t + u2

tt + 2utttux − u2x − 2uttuxt − 4ux uxtt + 2uxtuxx

− u2xx + 2ux uxxt + ut

( − uttt + 2ux + uxtt + uxxt − uxxx)

+ u(2utt + utttt − 4uxt − 3uxttt + 2uxx + 3uxxtt − uxxxt

)).

The multiplier,

Q2B = uxxx − uttt − 3utxx + 3uttx + ex , (15)

leads to the conserved vector,

�t2B = 1

2

( − u2t + u2

tt + utttux − u2x − 2uttuxt − ux uxtt + 2uxtuxx − u2

xx

− ux uxxt + ux uxxx − 2ut(ex − ux + uxtt − 2uxxt + uxxx

)+ u

(2utt − 4uxt + uxttt + 2uxx − 3uxxtt + 3uxxxt − uxxxx

)),

�x2B = 1

2

(u2

t − u2tt + 2ex ux − 2utttux + u2

x + 2uttuxt + 4ux uxtt − 2uxtuxx

+ u2xx − 2ux uxxt + ut

(uttt − 2ux − uxtt − uxxt + uxxx

)− u

(2ex + 2utt + utttt − 4uxt − 3uxttt + 2uxx + 3uxxtt − uxxxt

)).

2.1.3 The Gordon-type equation: uxx − utt − eu = 0. The multiplier,

Q3 = uxxx − uttt − 3utxx + 3uttx − 1

2(ux − ut )

3,

Pramana – J. Phys., Vol. 77, No. 3, September 2011 451

S Jamal and A H Kara

yields the conserved vector,

�t3 = 1

8

( − u4t + 4u2

tt + 3u3t ux + 4utttux − 4euu2

x + 8utt(eu − uxt

)− 16euuxt − 3uu2

x uxt − 4ux uxtt + 4uuxttt

− u2t

(4eu + 3u2

x + 3u(uxt − uxx

)) + 8euuxx + 3uu2x uxx

+ 8uxtuxx − 4u2xx − 4ux uxxt − 12uuxxtt + 4ux uxxx

+ ut(u3

x + ux(8eu + 6u

(uxt − uxx

)) − 8(uxtt − 2uxxt + uxxx

))+ 12uuxxxt − 4uuxxxx

),

�x3 = 1

8

( − 4u2tt − 4uutttt + u3

t ux − 8utttux + 4euu2x − u4

x

+ u2t

(4eu − 3u2

x + 3u(utt − uxt

)) + 16euuxt − 3uu2x uxt

+ utt( − 8eu + 3uu2

x + 8uxt) + 16ux uxtt + 12uuxttt − 8euuxx

− 8uxtuxx + 4u2xx − 8ux uxxt − 12uuxxtt

+ ut(4uttt + 3u3

x + ux( − 8eu − 6u

(utt − uxt

))− 4

(uxtt + uxxt − uxxx

)) + 4uuxxxt).

2.2 (1+2) Gordon-type equations

For the multi-dimensional Gordon-type eq. (3), higher-order symmetries and multipliersand the corresponding conserved quantities may be determined for k(u) = u only becausefor other forms of k(u) the underlying calculations produce negative results. This seemsto be a consequence of the underlying differential operator being linear only if k(u) = u(see Proposition 5.22 in [1]). In what follows we first assume a form of multiplier for theequation

uxx + uyy − utt − u = 0, (16)

and secondly we take a formal approach (the multiplier method) for finding multipliers ofthe equation.

From the multiplier (14), if we assume a multiplier of eq. (16) to be

QA = −uxxx − uyyy + uttt + 3utxx + 3utyy − 3uttx − 3utty,

we obtain the conserved vector,

�tA = 1

2

(u2

t − u2tt − utttuy + u2

y + 2uttuyt + uyuytt − 2uytuyy + u2yy

+ uyuyyt − uyuyyy − utttux + uyytux + u2x + 2uttuxt − 2uyyuxt

+ uxuxtt − ux uxyy − 2uytuxx + 2uyyuxx − 2uxtuxx + u2xx

+ uyuxxt + ux uxxt − uyuxxy − ux uxxx

+ ut( − 2uy + 2uytt − 4uyyt + 2uyyy − 2ux + 2uxtt + uxyy − 4uxxt

+ uxxy + 2uxxx)

− u(2utt − 4uyt + uyttt + 2uyy − 3uyytt + 3uyyyt − uyyyy − 4uxt

+ uxttt + 2uxyyt + 2uxx − 3uxxtt + 2uxxyt − 2uxxyy + 3uxxxt

− uxxxx))

,

452 Pramana – J. Phys., Vol. 77, No. 3, September 2011

Higher-order symmetries and conservation laws of Gordon-type equations

�xA = 1

2

( − u2t + u2

tt − uttuyy + 2utttux − 3uyttux + 2uyytux − uyyyux − u2x

− 2uttuxt + 2uyyuxt − 4ux uxtt + ux uxyy − uyyuxx + 2uxtuxx − u2xx

+ 2ux uxxt + ut( − uttt + uyyt + 2ux + uxtt − uxyy + uxxt − uxxx

)+ u

(2utt + utttt − uyytt − 4uxt − 3uxttt + 3uxytt − uxyyt + uxyyy

+ 2uxx + 3uxxtt − uxxyy − uxxxt))

,

�yA = 1

2

( − u2t + u2

tt + 2utttuy − u2y − 2uttuyt − 4uyuytt + 2uytuyy − u2

yy

+ 2uyuyyt − 3uyuxtt − uttuxx + 2uytuxx − uyyuxx + 2uyuxxt

+ ut( − uttt + 2uy + uytt + uyyt − uyyy + uxxt − uxxy

) + uyuxxy

− uyuxxx + u(2utt + utttt − 4uyt − 3uyttt + 2uyy + 3uyytt

− uyyyt + 3uxytt − uxxtt − uxxyt − uxxyy + uxxxy))

.

Similarly, if from the multiplier (15) we assume a multiplier of eq. (16) to be

QB = uxxx + uyyy − uttt − 3utxx − 3utyy + 3uttx + 3utty + ex + ey,

we obtain the conserved vector,

�tB = 1

2

( − u2t + u2

tt + utttuy − u2y − 2uttuyt − uyuytt + 2uytuyy − u2

yy

− uyuyyt + uyuyyy + utttux − uyytux − u2x − 2uttuxt + 2uyyuxt

− uxuxtt + ux uxyy + 2uytuxx − 2uyyuxx + 2uxtuxx − u2xx − uyuxxt

− ux uxxt + uyuxxy + ux uxxx

− ut(2ex + 2ey − 2uy + 2uytt − 4uyyt + 2uyyy − 2ux + 2uxtt + uxyy

− 4uxxt + uxxy + 2uxxx)

+ u(2utt − 4uyt + uyttt + 2uyy − 3uyytt + 3uyyyt − uyyyy − 4uxt

+ uxttt + 2uxyyt + 2uxx − 3uxxtt + 2uxxyt − 2uxxyy + 3uxxxt

− uxxxx))

,

�xB = 1

2

(u2

t − u2tt + uttuyy + 2ex ux + 2eyux − 2utttux + 3uyttux − 2uyytux

+ uyyyux + u2x + 2uttuxt − 2uyyuxt + 4ux uxtt − ux uxyy + uyyuxx

− 2uxtuxx + u2xx − 2ux uxxt

+ ut(uttt − uyyt − 2ux − uxtt + uxyy − uxxt + uxxx

)+ u

( − 2ex − 2utt − utttt + uyytt + 4uxt + 3uxttt − 3uxytt + uxyyt

− uxyyy − 2uxx − 3uxxtt + uxxyy + uxxxt))

,

�yB = 1

2

(u2

t − u2tt + 2ex uy + 2eyuy − 2utttuy + u2

y + 2uttuyt + 4uyuytt

− 2uytuyy + u2yy − 2uyuyyt + 3uyuxtt + uttuxx − 2uytuxx + uyyuxx

− 2uyuxxt − uyuxxy

+ ut(uttt − 2uy − uytt − uyyt + uyyy − uxxt + uxxy

) + uyuxxx

+ u( − 2ey − 2utt − utttt + 4uyt + 3uyttt − 2uyy − 3uyytt + uyyyt

− 3uxytt + uxxtt + uxxyt + uxxyy − uxxxy))

.

Pramana – J. Phys., Vol. 77, No. 3, September 2011 453

S Jamal and A H Kara

More formally, suppose

δ

δu

[Q

(uxx + uyy − utt − u

)] = 0, (17)

where Q = Q(x, y, t, ux , ux , uxx, uxy, uxxx, uxxy, uxyy, uyyy). Although not pursued here,the calculations may include derivatives of u with respect to t . Then,

Q[(

uxx + uyy − utt − u)] = Dt�

t + Dx�x + Dy�

y,

where �t ,�x ,�y is the conserved flow (�t being the conserved density).We obtain the following for the multiplier Q:

Q = 1

6

{−3

(−1

3qC4x3 + (aC4 + pyC4 + pC3 + qC1)x2

+ (−ny2C4 + ((−b + c) C4 − 2C1 p − 2nC3)y

− 4pC5 − 2aC1 + (−b + c) C3 − 2nC6

+ (−2C6 + 2C2)q − 2βC11)x

+ 1

3my3C4 + (nC1 − aC4 + mC3)y2

+ ((2C6 − 2C2)p + (b − c)C1 − 2aC3 + 4nC5

+ 2mC6 + 2C11α) y − 2pC8 − 2nC7 − 2aC2 − 2qC10

+ (−2c + 2b) C5 − 2αC13 − 2βC12 − 2mC9

)}, (18)

where Ci , i = 1, 2, 3, . . . , 13, are arbitrary constants and

α = ux , β = uy,

a = uxy, b = uxx, c = uyy,

m = uxxx, n = uxxy, p = uxyy, q = uyyy.

When we solve (18), we obtain the set of multipliers Qi together with their conserveddensities �t

i .

Q1 = ux,

�t1 = 1

2(−ut ux + uuxt),

�x1 = 1

2

(−u2 + u(−utt + uyy) + u2x

),

�y1 = 1

2(uyux − uuxy).

Q2 = uy,

�t2 = 1

2(−ut uy + uuyt),

�x2 = 1

2(uyux − uuxy),

�y2 = 1

2

(−u2 + u2y + u(−utt + uxx)

).

454 Pramana – J. Phys., Vol. 77, No. 3, September 2011

Higher-order symmetries and conservation laws of Gordon-type equations

Q3 = xuy − yux ,

�t3 = 1

2(ut (−xuy + yux ) + u(xuyt − yuxt)),

�x3 = 1

2(yu2 + ux (xuy − yux ) + u(yutt − uy − yuyy − xuxy)),

�y3 = 1

2(−xu2 + uy(xuy − yux ) + u(−xutt + ux + yuxy + xuxx)).

Q4 = uxxx,

�t4 = 1

2(−ut uxxx + uuxxxt),

�x4 = 1

2(u2

x + ux (uxtt − uxyy) + uxx(−utt + uyy + uxx)

+ u(−2uxx − uxxtt + uxxyy)),

�y4 = 1

2(uyuxxx − uuxxxy).

Q5 = uyyy,

�t5 = 1

2(−ut uyyy + uuyyyt),

�x5 = 1

2(uyyyux − uuxyyy),

�y5 = 1

2

(u2

y + uyy(−utt + uyy + uxx) + uy(uytt − uxxy)

+u(−2uyy − uyytt + uxxyy)).

Q6 = uxyy,

�t6 = 1

2(−ut uxyy + uuxyyt),

�x6 = 1

6

(u2

y − uttuyy + u2yy + 3ux uxyy + uyyuxx + uy(uytt − uyyy − uxxy)

−u(2uyy + uyytt − uyyyy + 2uxxyy)),

�y6 = 1

6(uyttux − uyyyux − 4uuxy − 2uttuxy + 2uyyuxy − 2uuxytt − uuxyyy

+ 2uxyuxx − ux uxxy + uy(2ux + uxtt + 2uxyy − uxxx) + 2uuxxxy).

Q7 = uxxy,

�t7 = 1

2(−ut uxxy + uuxxyt),

�x7 = 1

6(uyttux − uyyyux − 4uuxy − 2uttuxy + 2uyyuxy − 2uuxytt + 2uuxyyy

+ 2uxyuxx + 2ux uxxy + uy(2ux + uxtt − uxyy − uxxx) − uuxxxy),

�y7 = 1

6

(u2

x − uttuxx + uyyuxx + u2xx + 3uyuxxy + ux (uxtt − uxyy − uxxx)

−u(2uxx + uxxtt + 2uxxyy − uxxxx)).

Pramana – J. Phys., Vol. 77, No. 3, September 2011 455

S Jamal and A H Kara

Q8 = −uyx + xuyyy − yuxyy,

�t8 = 1

2(ut (−xuyyy + uxy + yuxyy) + u(xuyyyt − uxyt − yuxyyt)),

�x8 = 1

12

(−2yu2y − 2yuyuytt − uyuyy − 2yu2

yy + utt(uy + 2yuyy) + 2yuyuyyy

+ 6xuyyyux − 6ux uxy − 6yux uxyy − uyuxx − 2yuyyuxx + 2yuyuxxy

+ u(2uy + uytt + 4yuyy + 2yuyytt − 7uyyy − 2yuyyyy − 6xuxyyy

+5uxxy + 4yuxxyy))

,

�y8 = 1

12

(6xu2

y − 6xuttuyy + 6xu2yy + uttux − 2yuyttux − uyyux + 2yuyyyux

+ 4yuttuxy − 4yuyyuxy + 6xuyyuxx − ux uxx − 4yuxyuxx + 2yux uxxy

+ 2uy(3xuytt − 2yux − yuxtt − 3uxy − 2yuxyy − 3xuxxy + yuxxx)

+ u(−12xuyy − 6xuyytt + 2ux + uxtt + 8yuxy + 4yuxytt + 11uxyy

+2yuxyyy + 6xuxxyy − uxxx − 4yuxxxy)).

Q9 = uxx − 2xuxyy + 2yuxxy − uyy,

�t9 = 1

2(ut (uyy + 2xuxyy − uxx − 2yuxxy)

+ u(−uyyt − 2xuxyyt + uxxt2yuxxyt)),

�x9 = 1

6

(−2xu2y + 2xuttuyy − 2xu2

yy − uttux + 2yuyttux − 2uyyux

− 2yuyyyux − 4yuttuxy + 4yuyyuxy − 6xux uxyy − 2xuyyuxx

+ 4ux uxx + 4yuxyuxx + 4yux uxxy

− 2uy(xuytt − xuyyy − 2yux − yuxtt + yuxyy − xuxxy + yuxxx)

− u(−4xuyy − 2xuyytt + 2xuyyyy + 2ux + uxtt + 8yuxy + 4yuxytt

−10uxyy − 4yuxyyy − 4xuxxyy + 2uxxx + 2yuxxxy)),

�y9 = 1

6

(utt(uy + 4xuxy − 2yuxx)

− 2(xuyttux − xuyyyux − yu2

x − yuxuxtt + 2xuyyuxy + yux uxyy

− yuyyuxx + 2xuxyuxx − yu2xx − xux uxxy + yux uxxx

+ uy(2uyy+2xux + xuxtt+2xuxyy − uxx−3yuxxy − xuxxx))

+ u(2uy + uytt + 2(uyyy + 4xuxy + 2xuxytt + xuxyyy − 2yuxx

− yuxxtt −5uxxy − 2yuxxyy − 2xuxxxy + yuxxxx))).

Q10 = −xuxxy − xuyyy + yuxyy + yuxxx,

�t10 = 1

2(ut (xuyyy − yuxyy + xuxxy − yuxxx)

+ u(−xuyyyt + yuxyyt − xuxxyt + yuxxxt)),

456 Pramana – J. Phys., Vol. 77, No. 3, September 2011

Higher-order symmetries and conservation laws of Gordon-type equations

�x10 = 1

6

(yu2

y − yuttuyy + yu2yy − xuyttux − 2xuyyyux + 3yu2

x + 3yux uxtt

+ 2xuttuxy − 2xuyyuxy − 3yuttuxx + 4yuyyuxx − 2xuxyuxx

+ 3yu2xx − 2xux uxxy

+ uy(yuytt − yuyyy − 2xux − xuxtt + xuxyy − yuxxy + xuxxx)

+ u(−2yuyy − yuyytt + 3uyyy + yuyyyy + 4xuxy + 2xuxytt

+xuxyyy − 6yuxx − 3yuxxtt + 3uxxy + yuxxyy + xuxxxy)),

�y10 = 1

6

(−3xu2y + 3xuttuyy−3xu2

yy + yuyttux − yuyyyux − xu2x − xux uxtt

− 2yuttuxy+2yuyyuxy+xux uxyy+xuttuxx − 4xuyyuxx+2yuxyuxx

− xu2xx − yux uxxy + xux uxxx

− uy(3xuytt − y(2ux + uxtt + 2(uxyy + uxxx)))

− u(−6xuyy−3xuyytt+4yuxy+2yuxytt + 3uxyy + yuxyyy − 2xuxx

−xuxxtt + xuxxyy + 3uxxx + yuxxxy + xuxxxx)).

Q11 = −xuxx + x2uxyy + xuyy + y2uxxx − 2xyuyxx − 2yuxy,

�t11 = 1

2(−ut (xuyy − 2yuxy + x2uxyy − xuxx − 2xyuxxy + y2uxxx)

+ u(xuyyt − 2yuxyt + x2uxyyt − xuxxt − 2xyuxxyt + y2uxxxt)),

�x11 = 1

6

(−2u2 + x2u2y +x2uyuytt−yuyuyy + x2u2

yy − x2uyuyyy − 4xyuyux

− 2xyuyttux + 2xuyyux + 2xyuyyyux + 3y2u2x − 2xyuyuxtt

+ 3y2ux uxtt − 4xyuyyuxy − 6yux uxy + 2xyuyuxyy + 3x2ux uxyy

− 3y2ux uxyy − yuyuxx + x2uyyuxx + 3y2uyyuxx − 4xux uxx

− 4xyuxyuxx + 3y2u2xx + utt

(yuy − x2uyy + xux + 4xyuxy − 3y2uxx

)− x2uyuxxy − 4xyux uxxy + 2xyuyuxxx

+ u(−2utt + 2yuy + yuytt − uyy − 2x2uyy − x2uyytt − yuyyy

+ x2uyyyy + 2xux + xuxtt + 8xyuxy + 4xyuxytt − 10xuxyy

− 4xyuxyyy + 5uxx − 6y2uxx − 3y2uxxtt + 11yuxxy

−2x2uxxyy + 3y2uxxyy + 2xuxxx + 2xyuxxxy))

,

�y11 = 1

6

(4xuyuyy + 2x2uyux + x2uyttux − yuyyux − x2uyyyux − 2xyu2

x

+ x2uyuxtt − 2xyux uxtt − 6yuyuxy + 2x2uyyuxy + 2x2uyuxyy

+ 2xyux uxyy − 2xuyuxx − 2xyuyyuxx − yux uxx + 2x2uxyuxx

− 2xyu2xx + utt

(−xuy + yux − 2x2uxy + 2xyuxx) − 6xyuyuxxy

− x2ux uxxy − x2uyuxxx + 3y2uyuxxx + 2xyux uxxx

+ u(− 2xuy − xuytt − 2xuyyy + 2yux + yuxtt + 6uxy − 4x2uxy

− 2x2uxytt + 5yuxyy − x2uxyyy + 4xyuxx + 2xyuxxtt

+ 10xuxxy + 4xyuxxyy − 7yuxxx + 2x2uxxxy − 3y2uxxxy

− 2xyuxxxx))

.

Pramana – J. Phys., Vol. 77, No. 3, September 2011 457

S Jamal and A H Kara

Q12 = yuxx + x2uyyy − yuyy + y2uyxx − 2xyuyyx − 2xuxy,

�t12 = 1

2

(−ut(−yuyy + x2uyyy − 2xuxy − 2xyuxyy + yuxx + y2uxxy

)+ u

(−yuyyt + x2uyyyt − 2xuxyt − 2xyuxyyt + yuxxt + y2uxxyt))

,

�x12 = 1

6

(− 2xyu2y − 2xyuyuytt − xuyuyy − 2xyu2

yy + 2xyuyuyyy + 2y2uyux

+ y2uyttux − 2yuyyux + 3x2uyyyux − y2uyyyux + y2uyuxtt

+ 2y2uyyuxy − 6xux uxy + utt(xuy − y(−2xuyy + ux + 2yuxy))

− y2uyuxyy − 6xyux uxyy − xuyuxx − 2xyuyyuxx + 4yux uxx

+ 2y2uxyuxx + 2xyuyuxxy + 2y2ux uxxy − y2uyuxxx

+ u(2xuy + xuytt + 4xyuyy + 2xyuyytt − 7xuyyy − 2xyuyyyy

− 2yux − yuxtt + 6uxy − 4y2uxy − 2y2uxytt + 10yuxyy − 3x2uxyyy

+ 2y2uxyyy + 5xuxxy + 4xyuxxyy − 2yuxxx − y2uxxxy))

,

�y12 = 1

6

(−2u2 +3x2u2y +3x2uyuytt −4yuyuyy +3x2u2

yy −4xyuyux −xuyyux

+ 2xyuyyyux + y2u2x − 2xyuyuxtt + y2ux uxtt − 6xuyuxy − 4xyuyuxyy

− y2ux uxyy + 2yuyuxx + 3x2uyyuxx + y2uyyuxx − xux uxx

− 4xyuxyuxx + y2u2xx + utt

(yuy − 3x2uyy + xux + 4xyuxy − y2uxx

)− 3x2uyuxxy + 3y2uyuxxy + 2xyux uxxy + 2xyuyuxxx − y2ux uxxx

+ u(− 2utt + 2yuy + yuytt + 5uyy − 6x2uyy − 3x2uyytt + 2yuyyy

+ 2xux + xuxtt + 8xyuxy + 4xyuxytt + 11xuxyy + 2xyuxyyy

− uxx − 2y2uxx − y2uxxtt − 10yuxxy + 3x2uxxyy − 2y2uxxyy

−xuxxx − 4xyuxxxy + y2uxxxx) − 4xyuyyuxy − 2xyuyttux

).

Q13 = −xyuxx − 1

3x3uyyy + 1

3y3uxxx + xyuyy − y2uxy − xy2uxxy

+ yx2uxyy + x2uxy,

�t13 = 1

6

(ut

(− 3xyuyy + x3uyyy − 3x2uxy + 3y2uxy − 3x2 yuxyy + 3xyuxx

+ 3xy2uxxy − y3uxxx)

+ u(3xyuyyt − x3uyyyt + 3x2uxyt − 3y2uxyt

+ 3x2 yuxyyt − 3xyuxxt − 3xy2uxxyt + y3uxxxt))

,

�x13 = 1

12

(− 4yu2 + 2x2 yu2y + 2x2 yuyuytt + x2uyuyy − y2uyuyy + 2x2 yu2

yy

− 2x2 yuyuyyy − 4xy2uyux − 2xy2uyttux + 4xyuyyux − 2x3uyyyux

+ 2xy2uyyyux + 2y3u2x − 2xy2uyuxtt + 2y3ux uxtt − 4xy2uyyuxy

− 6y2ux uxy + 2xy2uyuxyy + 6x2 yux uxyy − 2y3ux uxyy + x2uyuxx

− y2uyuxx + 2x2 yuyyuxx + 2y3uyyuxx − 8xyux uxx − 4xy2uxyuxx

+2y3u2xx +utt

((−x2 + y2)

uy −2y(x2uyy −xux −2xyuxy + y2uxx

))− 2x2 yuyuxxy − 4xy2ux uxxy + 2xy2uyuxxx

+ u(− 4yutt − 2

(x2 − y2

)uy − x2uytt + y2uytt − 2yuyy − 4x2 yuyy

458 Pramana – J. Phys., Vol. 77, No. 3, September 2011

Higher-order symmetries and conservation laws of Gordon-type equations

− 2x2 yuyytt + 7x2uyyy − y2uyyy + 2x2 yuyyyy + 4xyux + 2xyuxtt

− 12xuxy + 8xy2uxy + 4xy2uxytt + 2x3uxyyy − 4xy2uxyyy

+ 10yuxx − 4y3uxx − 2y3uxxtt − 5x2uxxy − 4x2 yuxxyy + 2y3uxxyy

+ 4xyuxxx + 2xy2uxxxy − 20xyuxyy + 11y2uxxy) + 6x2ux uxy

),

�y13 = 1

12

(4xu2 − 2x3u2

y − 2x3uyuytt + 8xyuyuyy − 2x3u2yy + 4x2 yuyux

+ 2x2 yuyttux + x2uyyux − y2uyyux − 2x2 yuyyyux − 2xy2u2x

+ 2x2 yuyuxtt − 2xy2uxuxtt + 6x2uyuxy − 6y2uyuxy + 4x2 yuyyuxy

+ 4x2 yuyuxyy + 2xy2uxuxyy − 4xyuyuxx − 2x3uyyuxx− 2xy2uyyuxx

+ x2ux uxx − y2ux uxx + 4x2 yuxyuxx − 2xy2u2xx

+ utt(−2xyuy + 2x3uyy − x2ux + y2ux − 4x2 yuxy + 2xy2uxx

)+ 2x3uyuxxy − 6xy2uyuxxy − 2x2 yux uxxy − 2x2 yuyuxxx

+ 2y3uyuxxx + 2xy2ux uxxx

+ u(4xutt − 4xyuy − 2xyuytt − 10xuyy + 4x3uyy + 2x3uyytt

− 4xyuyyy − 2x2ux + 2y2ux − x2uxtt + 12yuxy − 8x2 yuxy

− 4x2 yuxytt − 11x2uxyy + 5y2uxyy − 2x2 yuxyyy + 4xy2uxx

+ 2xy2uxxtt + 20xyuxxy − 2x3uxxyy + 4xy2uxxyy + x2uxxx

− 7y2uxxx + 4x2 yuxxxy − 2y3uxxxy − 2xy2uxxxx + 2xuxx

+y2uxtt))

.

We now select a few of the symmetries above and prove that they are variational usingTheorem 1 where X = Qi∂u .

X E + AFQ4 E = Qxx + Qyy − Qtt − Q + (−Dxxx) (uxx + uyy − utt − u)

= uxxxxx + uxxxyy − uxxxtt − uxxx

− uxxxxx − uxxxyy + uxxxtt + uxxx = 0,

X E + AFQ8 E = Qxx + Qyy − Qtt − Q

+ (Dxy + y Dxyy − x Dyyy

)(uxx + uyy − utt − u)

= uxxxy + uxyyy − uxytt − uxy + y(uxxxyy + uxyyyy − uttxyy − uxyy)

− x(uxxyyy + uyyyyy − uttyyy − uyyy) + 2uxyyy

+ xuxxyyy − yuxxxyy − uxxxy + uyyyyy − 2uxyyy − yuxyyyy

− uxyyy − xuyyytt + yuxyytt + uxytt − xuyyy

+ yuxyy + uxy = 0,

X E + AFQ11 E = Qxx + Qyy − Qtt − Q

+ (x Dxx + 2y Dxy − x Dyy − y2 Dxxx + 2xy Dxxy

− x2 Dxyy)(uxx + uyy − utt − u)

= x (uxxxx + uxxyy − uxxtt − uxx)

+ 2y(uxxxy + uxyyy − uttxyy − uxy)

− x(uxxyy + uyyyyy − uttyy − uyy)

− y2(uxxxxx + uxxxyy − uxxxtt − uxxx)

Pramana – J. Phys., Vol. 77, No. 3, September 2011 459

S Jamal and A H Kara

+ 2xy(uxxxxy + uxxyyy − uttxxy − uxxy)

− x2(uxxxyy + uyyyyx − uxyytt − uxyy) + 4uxyy + 4xuxxyy

+ x2uxxxyy − 4yuxxxyy − 2xyuxxxxy + xuxxyy − xuxxxx

+ y2uxxxxx − 2yuxxxy + x2uxyyyy − 4xuxxyy − 2xyuxxyyy

− xuxxyy + 2yuxxxy + 2uxxx + 2yuxxxy + y2uxxxyy

− 4uxyy − 2yuxyyy − x2uxyytt + 2xyuxxytt − xuyytt + xuxxtt

− y2uxxxtt + 2yuxytt − 2uxxx − x2uxyy + 2xyuxxy − xuyy

+ xuxx − y2uxxx + 2yuxy + xuyyyy = 0.

Thus, the evolutionary, higher-order symmetries Xi = Qi∂u are variational.

3. Concluding remarks

We studied the classical forms of certain classes of (1+2) Gordon-type equations to deter-mine possible higher-order symmetries. These were obtained by investigating the classical(1+1) Gordon-type equations and, more formally, using the multiplier approach. It turnedout that only the (1+2) Klein–Gordon-type equation produced such symmetries which,in fact, were proved to be variational. The corresponding conserved densities were alsocalculated.

References

[1] P Olver, Applications of Lie groups to differential equations (Springer, New York, 1993)[2] S Anco and G Bluman, Eur. J. Appl. Math. 13, 545 (2002)[3] G Bluman and S Kumei, Symmetries and differential equations (Springer-Verlag, New York,

1989)[4] W Hereman, Int. J. Quant. Chem. 106, 278 (2006)[5] U Göktas and W Hereman, Physica D123, 425 (1998)[6] N H Ibragimov, A H Kara and F M Mahomed, Nonlin. Dyn. 15, 115 (1998)[7] A H Kara and F M Mahomed, Int. J. Theoret. Phys. 39, 23 (2000)[8] A H Kara and F M Mahomed, J. Nonlin. Math. Phys. 9, 60 (2002)[9] S Jamal and A H Kara, to appear in Nonlinear Dynamics, DOI: 10.1007/s11071-011-9961-1

(2011)

460 Pramana – J. Phys., Vol. 77, No. 3, September 2011