Higher-order symmetries and conservation laws of multi-dimensional

PRAMANA

c Indian Academy of Sciences

— journal of physics

Vol. 77, No. 3 September 2011 pp. 447–460

Higher-order symmetries and conservation laws of multi-dimensional Gordon-type equations S JAMAL1,∗ and A H KARA2 1 School

of Mathematics, University of the Witwatersrand, Johannesburg, Private Bag 3, Wits 2050, South Africa 2 School of Mathematics and Centre for Differential Equations, Continuum Mechanics and Applications, University of the Witwatersrand, Johannesburg, Private Bag 3, Wits 2050, South Africa ∗ Corresponding author. E-mail: [email protected] Abstract. In this paper a class of multi-dimensional Gordon-type equations are analysed using a multiplier and homotopy approach to construct conservation laws. The main focus is the analysis of the classical versions of the Gordon-type equations and obtaining higher-order variational symmetries and corresponding conserved quantities. The results are extended to the multi-dimensional Gordontype equations with the two-dimensional Klein–Gordon equation in particular yielding interesting results. 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 Gordontype equations, u X T − k(u) = 0,

(1)

can be recast in the well-known classical form, u tt − u xx − k(u) = 0.

(2)

The purpose for doing this is to gain an insight into extending the results to the multidimensional Gordon-type equations which are somewhat cumbersome if one was to pursue a canonical form. That is, the multi-dimensional Gordon-type equations are best considered as u tt − u − k(u) = 0, DOI: 10.1007/s12043-011-0165-5; ePublication: 26 August 2011

(3) 447

S Jamal and A H Kara where denotes the Laplacian. This equation is of the classical form rather than the canonical form. Equation (1) has been extensively studied in terms of their symmetries and variational properties [1]. In particular, the sine-Gordon equation u X T − sin u = 0 has been shown to have higher-order variational symmetries, X = Q∂u . For example, 1 3 X 1 = u X X X + u X ∂u , 2 5 2 5 3 5 2 X 2 = u X X X X X + u X u X X X + u X u X X + u X ∂u , 2 2 8 1 3 X 3 = u T T T + u T ∂u . 2

(4)

Of the variational symmetries in (4), the first two lead to the corresponding higher-order conserved densities, 1 1 1T = − u 2X X + u 4X , 2 8

2T =

1 2 5 1 u X X X − u 2X u 2X X + u 6X . 2 4 16

(5)

In a similar way, we investigate the existence of higher-order variational symmetries for other classes of Gordon-type equations with dimension one and two, and we find possible conserved densities via the multiplier method outlined below. We conclude with some interesting and unexpected results. We apply the multiplier approach that leads to a large class of interesting and higherorder conserved flows that would not have been obtained by variational techniques such as Noether’s theorem. In particular, we obtain higher-order multipliers. We present some of the definitions and notations below. Intrinsic to a Lie algebraic treatment of differential equations is the universal space A (see [1]). The space A is the 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 variables x = (x 1 , x 2 , . . . , x n ) and m dependent variables u = (u 1 , u 2 , . . . , u m ), G μ (x, u, u (1) , . . . , u (r ) ) = 0,

μ = 1, . . . , m, ˜

(6)

where u (1) , u (2) , . . . , u (r ) denote the collections of all first-, second-, . . ., r th-order partial derivatives, that is, u iα = Di (u α ), u iαj = D j Di (u α ), . . . respectively, with the total differentiation operator with respect to x i given by Di =

∂ ∂ ∂ + u iα α + u iαj α + · · · , ∂xi ∂u ∂u j

i = 1, . . . , n,

(7)

where the summation convention is used whenever appropriate. A current = (1 , . . . , n ) is conserved if it satisfies Di i = 0 448

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

(8)

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 from multipliers Q μ (x, u, u (1) , . . .) such that Q μ G μ = Di i

(9)

holds identically (that is, off the solution space) for some current . The conserved vector may then be obtained by the homotopy operator (see [2–4]). Other works on symmetries and conservation laws can be found in [5–8]. Definition 1. The Euler operator for each dependent variable u α is defined by δ ∂ ∂ = α + (−1)s Di1 · · · Dis , α α δu ∂u ∂u i1 ···is s≥1

α = 1, . . . , m.

(10)

In most literature, a variational problem consists of finding the extrema (maxima or minima) 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 connected subset with smooth boundary ∂ (we consider the Euclidean space with X = R n ). The integrand L(x, u (n) ), called the Lagrangian of the variational problem L, is a smooth function of x, u and various derivatives of u [1]. Definition 2. The Lie–Bäcklund or generalized operator is given by X = ξi

∂ ∂ + ηα α , i ∂x ∂u

ξ i , ηα ∈ A.

(11)

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

∂ ∂ ∂ + ηα α + ζiα1 ...is , α i ∂x ∂u ∂u i1 ...is s≥1

(12)

where the additional coefficients are determined uniquely by the prolongation formulae, ζiα = Di (W α ) + ξ j u iαj , ζiα1 ...is = Di1 . . . Dis (W α ) + ξ j u αji1 ...is ,

s > 1.

(13)

Definition 3. A Lie–Bäcklund operator X of the form (11) is called a variational symmetry if it leaves invariant the functional L[u] = L(x, u (n) )dx. Variational partial differential 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, an evolutionary vector field X = Q∂u is a variational symmetry if and only if X E + AF Q E = 0, where AFQ is the adjoint Fréchet derivative on Q [1]. Pramana – J. Phys., Vol. 77, No. 3, September 2011

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S Jamal and A H Kara 2. Applications We now construct symmetries and conservation laws for the classes of eq. (2) discussed above with particular emphasis on the main eq. (3). Most of the tedious calculations have been 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 one or two of the higher-order multipliers and conserved densities which arise as a consequence of the transformation to the classical form. Other multipliers and conserved densities exist and may be found in [9]. 2.1.1 The Gordon-type equation: u xx − u tt − sin u = 0. In converting the equation u xx − u tt − sin(u) = 0 into u X T − sin u, we use the transformations, 1 1 T = (x + t). X = (x − t), 2 2 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 X¯ . We shall now illustrate the method using X1 = (u X X X + 12 u 3X )∂u listed in (4). Since u X = u x x X + ut tX = u x − ut , u X X = u xx x X + u xt t X − u tx x X − u tt t X = u xx − 2u xt + u tt , u X X X = u xxx x X + u xxt t X − 2(u xtx x X + u xtt t X ) + u ttx x X + u ttt t X = u xxx − 3u xxt + 3u xtt − u ttt , the equivalent of X1 is 1 3 ¯ X1 = u xxx − 3u xxt + 3u xtt − u ttt + (u x − u t ) ∂u , 2 with the multiplier, 1 Q1 = (u x − u t )3 + u xxx − u ttt − 3u txx + 3u ttx , 2 leading to a conserved vector, 1 4 u + 4u 2tt − 3u 3t u x + 4u ttt u x − 4 cos(u) u 2x + 8u tt (sin(u) − u xt ) t1 = 8 t − 16 sin(u) u xt + 3uu 2x u xt − 4u x u xtt + 4uu xttt + u 2t −4 cos(u) + 3u 2x + 3u (u xt − u xx ) + 8 sin(u) u xx − 3uu 2x u xx + 8u xt u xx − 4u 2xx − 4u x u xxt − 12uu xxtt + 4u x u xxx + u t −u 3x + u x (8 cos(u) − 6u(u xt − u xx )) − 8 (u xtt − 2u xxt + u xxx ) + 12uu xxxt − 4uu xxxx , 450


Higher-order symmetries and conservation laws of Gordon-type equations 1x =

1 −4u 2tt − 4uu tttt − u 3t u x − 8u ttt u x + 4 cos uu 2x + u 4x 8 + u 2t 4 cos u + 3u 2x − 3u (u tt − u xt ) + 16 sin uu xt + 3uu 2x u xt + u tt −8 sin u − 3uu 2x + 8u xt + 16u x u xtt + 12uu xttt − 8 sin uu xx − 8u xt u xx + 4u 2xx − 8u x u xxt − 12uu xxtt + u t 4u ttt − 3u 3x + u x (−8 cos u + 6u (u tt − u xt )) − 4 (u xtt + u xxt − u xxx )) + 4uu xxxt .

2.1.2 The Gordon-type equation: u xx − u tt − u = 0. The multiplier, Q2A = −u xxx + u ttt + 3u txx − 3u ttx ,

(14)

leads to the conserved vector, 1 2 u − u 2tt − u ttt u x + u 2x + 2u tt u xt + u x u xtt − 2u xt u xx + u 2xx 2 t + u x u xxt − 2u t u x − u xtt + 2u xxt − u xxx − u x u xxx + u − 2u tt + 4u xt − u xttt − 2u xx + 3u xxtt − 3u xxxt + u xxxx , 1 − u 2t + u 2tt + 2u ttt u x − u 2x − 2u tt u xt − 4u x u xtt + 2u xt u xx = 2 − u 2xx + 2u x u xxt + u t − u ttt + 2u x + u xtt + u xxt − u xxx + u 2u tt + u tttt − 4u xt − 3u xttt + 2u xx + 3u xxtt − u xxxt .

t2A =

x 2A

The multiplier, Q2B = u xxx − u ttt − 3u txx + 3u ttx + e x ,

(15)

leads to the conserved vector, 1 − u 2t + u 2tt + u ttt u x − u 2x − 2u tt u xt − u x u xtt + 2u xt u xx − u 2xx 2 − u x u xxt + u x u xxx − 2u t e x − u x + u xtt − 2u xxt + u xxx + u 2u tt − 4u xt + u xttt + 2u xx − 3u xxtt + 3u xxxt − u xxxx , 1 2 u − u 2tt + 2e x u x − 2u ttt u x + u 2x + 2u tt u xt + 4u x u xtt − 2u xt u xx = 2 t + u 2xx − 2u x u xxt + u t u ttt − 2u x − u xtt − u xxt + u xxx x − u 2e + 2u tt + u tttt − 4u xt − 3u xttt + 2u xx + 3u xxtt − u xxxt .

t2B =

x 2B

2.1.3 The Gordon-type equation: u xx − u tt − eu = 0. The multiplier, 1 Q3 = u xxx − u ttt − 3u txx + 3u ttx − (u x − u t )3 , 2 Pramana – J. Phys., Vol. 77, No. 3, September 2011

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S Jamal and A H Kara yields the conserved vector, 1 − u 4t + 4u 2tt + 3u 3t u x + 4u ttt u x − 4eu u 2x + 8u tt eu − u xt t3 = 8 − 16eu u xt − 3uu 2x u xt − 4u x u xtt + 4uu xttt − u 2t 4eu + 3u 2x + 3u u xt − u xx + 8eu u xx + 3uu 2x u xx + 8u xt u xx − 4u 2xx − 4u x u xxt − 12uu xxtt + 4u x u xxx + u t u 3x + u x 8eu + 6u u xt − u xx − 8 u xtt − 2u xxt + u xxx + 12uu xxxt − 4uu xxxx , 1 − 4u 2tt − 4uu tttt + u 3t u x − 8u ttt u x + 4eu u 2x − u 4x 3x = 8 + u 2t 4eu − 3u 2x + 3u u tt − u xt + 16eu u xt − 3uu 2x u xt + u tt − 8eu + 3uu 2x + 8u xt + 16u x u xtt + 12uu xttt − 8eu u xx − 8u xt u xx + 4u 2xx − 8u x u xxt − 12uu xxtt + u t 4u ttt + 3u 3x + u x − 8eu − 6u u tt − u xt − 4 u xtt + u xxt − u xxx + 4uu xxxt . 2.2 (1+2) Gordon-type equations For the multi-dimensional Gordon-type eq. (3), higher-order symmetries and multipliers and the corresponding conserved quantities may be determined for k(u) = u only because for other forms of k(u) the underlying calculations produce negative results. This seems to 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 the equation u xx + u yy − u tt − u = 0,

(16)

and secondly we take a formal approach (the multiplier method) for finding multipliers of the equation. From the multiplier (14), if we assume a multiplier of eq. (16) to be Q A = −u xxx − u yyy + u ttt + 3u txx + 3u tyy − 3u ttx − 3u tty , we obtain the conserved vector, 1 2 tA = u − u 2tt − u ttt u y + u 2y + 2u tt u yt + u y u ytt − 2u yt u yy + u 2yy 2 t + u y u yyt − u y u yyy − u ttt u x + u yyt u x + u 2x + 2u tt u xt − 2u yy u xt + u x u xtt − u x u xyy − 2u yt u xx + 2u yy u xx − 2u xt u xx + u 2xx + u y u xxt + u x u xxt − u y u xxy − u x u xxx + u t − 2u y + 2u ytt − 4u yyt + 2u yyy − 2u x + 2u xtt + u xyy − 4u xxt + u xxy + 2u xxx − u 2u tt − 4u yt + u yttt + 2u yy − 3u yytt + 3u yyyt − u yyyy − 4u xt + u xttt + 2u xyyt + 2u xx − 3u xxtt + 2u xxyt − 2u xxyy + 3u xxxt − u xxxx , 452


Higher-order symmetries and conservation laws of Gordon-type equations 1 − u 2t + u 2tt − u tt u yy + 2u ttt u x − 3u ytt u x + 2u yyt u x − u yyy u x − u 2x 2 − 2u tt u xt + 2u yy u xt − 4u x u xtt + u x u xyy − u yy u xx + 2u xt u xx − u 2xx + 2u x u xxt + u t − u ttt + u yyt + 2u x + u xtt − u xyy + u xxt − u xxx + u 2u tt + u tttt − u yytt − 4u xt − 3u xttt + 3u xytt − u xyyt + u xyyy + 2u xx + 3u xxtt − u xxyy − u xxxt , 1 y − u 2t + u 2tt + 2u ttt u y − u 2y − 2u tt u yt − 4u y u ytt + 2u yt u yy − u 2yy A = 2 + 2u y u yyt − 3u y u xtt − u tt u xx + 2u yt u xx − u yy u xx + 2u y u xxt + u t − u ttt + 2u y + u ytt + u yyt − u yyy + u xxt − u xxy + u y u xxy − u y u xxx + u 2u tt + u tttt − 4u yt − 3u yttt + 2u yy + 3u yytt − u yyyt + 3u xytt − u xxtt − u xxyt − u xxyy + u xxxy .

xA =

Similarly, if from the multiplier (15) we assume a multiplier of eq. (16) to be Q B = u xxx + u yyy − u ttt − 3u txx − 3u tyy + 3u ttx + 3u tty + e x + e y , we obtain the conserved vector, 1 − u 2t + u 2tt + u ttt u y − u 2y − 2u tt u yt − u y u ytt + 2u yt u yy − u 2yy 2 − u y u yyt + u y u yyy + u ttt u x − u yyt u x − u 2x − 2u tt u xt + 2u yy u xt − u x u xtt + u x u xyy + 2u yt u xx − 2u yy u xx + 2u xt u xx − u 2xx − u y u xxt − u x u xxt + u y u xxy + u x u xxx − u t 2e x + 2e y − 2u y + 2u ytt − 4u yyt + 2u yyy − 2u x + 2u xtt + u xyy − 4u xxt + u xxy + 2u xxx + u 2u tt − 4u yt + u yttt + 2u yy − 3u yytt + 3u yyyt − u yyyy − 4u xt + u xttt + 2u xyyt + 2u xx − 3u xxtt + 2u xxyt − 2u xxyy + 3u xxxt − u xxxx , 1 2 u − u 2tt + u tt u yy + 2e x u x + 2e y u x − 2u ttt u x + 3u ytt u x − 2u yyt u x xB = 2 t + u yyy u x + u 2x + 2u tt u xt − 2u yy u xt + 4u x u xtt − u x u xyy + u yy u xx − 2u xt u xx + u 2xx − 2u x u xxt + u t u ttt − u yyt − 2u x − u xtt + u xyy − u xxt + u xxx + u − 2e x − 2u tt − u tttt + u yytt + 4u xt + 3u xttt − 3u xytt + u xyyt − u xyyy − 2u xx − 3u xxtt + u xxyy + u xxxt , 1 2 y u − u 2tt + 2e x u y + 2e y u y − 2u ttt u y + u 2y + 2u tt u yt + 4u y u ytt B = 2 t − 2u yt u yy + u 2yy − 2u y u yyt + 3u y u xtt + u tt u xx − 2u yt u xx + u yy u xx − 2u y u xxt − u y u xxy + u t u ttt − 2u y − u ytt − u yyt + u yyy − u xxt + u xxy + u y u xxx + u − 2e y − 2u tt − u tttt + 4u yt + 3u yttt − 2u yy − 3u yytt + u yyyt − 3u xytt + u xxtt + u xxyt + u xxyy − u xxxy . tB =


453

S Jamal and A H Kara More formally, suppose

δ Q u xx + u yy − u tt − u = 0, δu

(17)

where Q = Q(x, y, t, u x , u x , u xx , u xy , u xxx , u xxy , u xyy , u yyy ). Although not pursued here, the calculations may include derivatives of u with respect to t. Then,

Q u xx + u yy − u tt − u = Dt t + Dx x + D y y , where t , x , y is the conserved flow (t being the conserved density). We obtain the following for the multiplier Q: 1 1 Q= −3 − qC4 x 3 + (aC4 + pyC4 + pC3 + qC1 )x 2 6 3 + −ny 2 C4 + ((−b + c) C4 − 2C1 p − 2nC3 )y − 4 pC5 − 2aC1 + (−b + c) C3 − 2nC6 + (−2C6 + 2C2 )q − 2βC11 x 1 + my 3 C4 + (nC1 − aC4 + mC3 )y 2 3 + ((2C6 − 2C2 ) p + (b − c)C1 − 2aC3 + 4nC5 + 2mC6 + 2C11 α) y − 2 pC8 − 2nC7 − 2aC2 − 2qC10 + (−2c + 2b) C5 − 2αC13 − 2βC12 − 2mC9 , (18) where Ci , i = 1, 2, 3, . . . , 13, are arbitrary constants and α = ux , β = u y, a = u xy , b = u xx , c = u yy , m = u xxx , n = u xxy , p = u xyy , q = u yyy . When we solve (18), we obtain the set of multipliers Qi together with their conserved densities it . Q1 = u x , 1 t1 = (−u t u x + uu xt ), 2 1 2 x −u + u(−u tt + u yy ) + u 2x , 1 = 2 1 y 1 = (u y u x − uu xy ). 2 Q2 = u y , 1 t2 = (−u t u y + uu yt ), 2 1 2x = (u y u x − uu xy ), 2 1 2 y 2 = −u + u 2y + u(−u tt + u xx ) . 2 454


Higher-order symmetries and conservation laws of Gordon-type equations Q3 = xu y − yu x , 1 t3 = (u t (−xu y + yu x ) + u(xu yt − yu xt )), 2 1 x 3 = (yu 2 + u x (xu y − yu x ) + u(yu tt − u y − yu yy − xu xy )), 2 1 y 3 = (−xu 2 + u y (xu y − yu x ) + u(−xu tt + u x + yu xy + xu xx )). 2 Q4 = u xxx , 1 t4 = (−u t u xxx + uu xxxt ), 2 1 4x = (u 2x + u x (u xtt − u xyy ) + u xx (−u tt + u yy + u xx ) 2 + u(−2u xx − u xxtt + u xxyy )), 1 y 4 = (u y u xxx − uu xxxy ). 2 Q5 = u yyy , 1 t5 = (−u t u yyy + uu yyyt ), 2 1 x 5 = (u yyy u x − uu xyyy ), 2 1 2 y u + u yy (−u tt + u yy + u xx ) + u y (u ytt − u xxy ) 5 = 2 y +u(−2u yy − u yytt + u xxyy ) . Q6 = u xyy , 1 t6 = (−u t u xyy + uu xyyt ), 2 1 2 x u − u tt u yy + u 2yy + 3u x u xyy + u yy u xx + u y (u ytt − u yyy − u xxy ) 6 = 6 y −u(2u yy + u yytt − u yyyy + 2u xxyy ) , 1 y 6 = (u ytt u x − u yyy u x − 4uu xy − 2u tt u xy + 2u yy u xy − 2uu xytt − uu xyyy 6 + 2u xy u xx − u x u xxy + u y (2u x + u xtt + 2u xyy − u xxx ) + 2uu xxxy ). Q7 = u xxy , 1 t7 = (−u t u xxy + uu xxyt ), 2 1 x 7 = (u ytt u x − u yyy u x − 4uu xy − 2u tt u xy + 2u yy u xy − 2uu xytt + 2uu xyyy 6 + 2u xy u xx + 2u x u xxy + u y (2u x + u xtt − u xyy − u xxx ) − uu xxxy ), 2 1 y u − u tt u xx + u yy u xx + u 2xx + 3u y u xxy + u x (u xtt − u xyy − u xxx ) 7 = 6 x −u(2u xx + u xxtt + 2u xxyy − u xxxx ) . Pramana – J. Phys., Vol. 77, No. 3, September 2011

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S Jamal and A H Kara Q8 = −u yx + xu yyy − yu xyy , 1 (u t (−xu yyy + u xy + yu xyy ) + u(xu yyyt − u xyt − yu xyyt )), 2 1 −2yu 2y − 2yu y u ytt − u y u yy − 2yu 2yy + u tt (u y + 2yu yy ) + 2yu y u yyy 8x = 12 + 6xu yyy u x − 6u x u xy − 6yu x u xyy − u y u xx − 2yu yy u xx + 2yu y u xxy + u 2u y + u ytt + 4yu yy + 2yu yytt − 7u yyy − 2yu yyyy − 6xu xyyy +5u xxy + 4yu xxyy , t8 =

y

8 =

1 6xu 2y − 6xu tt u yy + 6xu 2yy + u tt u x − 2yu ytt u x − u yy u x + 2yu yyy u x 12 + 4yu tt u xy − 4yu yy u xy + 6xu yy u xx − u x u xx − 4yu xy u xx + 2yu x u xxy + 2u y (3xu ytt − 2yu x − yu xtt − 3u xy − 2yu xyy − 3xu xxy + yu xxx ) + u(−12xu yy − 6xu yytt + 2u x + u xtt + 8yu xy + 4yu xytt + 11u xyy +2yu xyyy + 6xu xxyy − u xxx − 4yu xxxy ) .

Q9 = u xx − 2xu xyy + 2yu xxy − u yy , t9 =

9x =

y

9 =

1 (u t (u yy + 2xu xyy − u xx − 2yu xxy ) 2 + u(−u yyt − 2xu xyyt + u xxt 2yu xxyt )), 1 −2xu 2y + 2xu tt u yy − 2xu 2yy − u tt u x + 2yu ytt u x − 2u yy u x 6 − 2yu yyy u x − 4yu tt u xy + 4yu yy u xy − 6xu x u xyy − 2xu yy u xx + 4u x u xx + 4yu xy u xx + 4yu x u xxy − 2u y (xu ytt − xu yyy − 2yu x − yu xtt + yu xyy − xu xxy + yu xxx ) − u(−4xu yy − 2xu yytt + 2xu yyyy + 2u x + u xtt + 8yu xy + 4yu xytt −10u xyy − 4yu xyyy − 4xu xxyy + 2u xxx + 2yu xxxy ) , 1 u tt (u y + 4xu xy − 2yu xx ) 6 − 2 xu ytt u x − xu yyy u x − yu 2x − yu x u xtt + 2xu yy u xy + yu x u xyy − yu yy u xx + 2xu xy u xx − yu 2xx − xu x u xxy + yu x u xxx + u y (2u yy +2xu x + xu xtt +2xu xyy − u xx −3yu xxy − xu xxx ) + u(2u y + u ytt + 2(u yyy + 4xu xy + 2xu xytt + xu xyyy − 2yu xx − yu xxtt −5u xxy − 2yu xxyy − 2xu xxxy + yu xxxx )) .

Q10 = −xu xxy − xu yyy + yu xyy + yu xxx , t10 =

456

1 (u t (xu yyy − yu xyy + xu xxy − yu xxx ) 2 + u(−xu yyyt + yu xyyt − xu xxyt + yu xxxt )),


Higher-order symmetries and conservation laws of Gordon-type equations 1 2 yu y − yu tt u yy + yu 2yy − xu ytt u x − 2xu yyy u x + 3yu 2x + 3yu x u xtt 6 + 2xu tt u xy − 2xu yy u xy − 3yu tt u xx + 4yu yy u xx − 2xu xy u xx + 3yu 2xx − 2xu x u xxy + u y (yu ytt − yu yyy − 2xu x − xu xtt + xu xyy − yu xxy + xu xxx ) + u(−2yu yy − yu yytt + 3u yyy + yu yyyy + 4xu xy + 2xu xytt +xu xyyy − 6yu xx − 3yu xxtt + 3u xxy + yu xxyy + xu xxxy ) , 1 −3xu 2y + 3xu tt u yy −3xu 2yy + yu ytt u x − yu yyy u x − xu 2x − xu x u xtt = 6 − 2yu tt u xy +2yu yy u xy +xu x u xyy +xu tt u xx − 4xu yy u xx +2yu xy u xx − xu 2xx − yu x u xxy + xu x u xxx − u y (3xu ytt − y(2u x + u xtt + 2(u xyy + u xxx ))) − u(−6xu yy −3xu yytt +4yu xy +2yu xytt + 3u xyy + yu xyyy − 2xu xx −xu xxtt + xu xxyy + 3u xxx + yu xxxy + xu xxxx ) .

x = 10

y

10

Q11 = −xu xx + x 2 u xyy + xu yy + y 2 u xxx − 2x yu yxx − 2yu xy , 1 t11 = (−u t (xu yy − 2yu xy + x 2 u xyy − xu xx − 2x yu xxy + y 2 u xxx ) 2 + u(xu yyt − 2yu xyt + x 2 u xyyt − xu xxt − 2x yu xxyt + y 2 u xxxt )), 1 x −2u 2 + x 2 u 2y +x 2 u y u ytt − yu y u yy + x 2 u 2yy − x 2 u y u yyy − 4x yu y u x = 11 6 − 2x yu ytt u x + 2xu yy u x + 2x yu yyy u x + 3y 2 u 2x − 2x yu y u xtt + 3y 2 u x u xtt − 4x yu yy u xy − 6yu x u xy + 2x yu y u xyy + 3x 2 u x u xyy − 3y 2 u x u xyy − yu y u xx + x 2 u yy u xx + 3y 2 u yy u xx − 4xu x u xx − 4x yu xy u xx + 3y 2 u 2xx + u tt yu y − x 2 u yy + xu x + 4x yu xy − 3y 2 u xx − x 2 u y u xxy − 4x yu x u xxy + 2x yu y u xxx + u −2u tt + 2yu y + yu ytt − u yy − 2x 2 u yy − x 2 u yytt − yu yyy + x 2 u yyyy + 2xu x + xu xtt + 8x yu xy + 4x yu xytt − 10xu xyy − 4x yu xyyy + 5u xx − 6y 2 u xx − 3y 2 u xxtt + 11yu xxy −2x 2 u xxyy + 3y 2 u xxyy + 2xu xxx + 2x yu xxxy ,

y

11 =

1 4xu y u yy + 2x 2 u y u x + x 2 u ytt u x − yu yy u x − x 2 u yyy u x − 2x yu 2x 6 + x 2 u y u xtt − 2x yu x u xtt − 6yu y u xy + 2x 2 u yy u xy + 2x 2 u y u xyy + 2x yu x u xyy − 2xu y u xx − 2x yu yy u xx − yu x u xx + 2x 2 u xy u xx − 2x yu 2xx + u tt −xu y + yu x − 2x 2 u xy + 2x yu xx − 6x yu y u xxy − x 2 u x u xxy − x 2 u y u xxx + 3y 2 u y u xxx + 2x yu x u xxx + u − 2xu y − xu ytt − 2xu yyy + 2yu x + yu xtt + 6u xy − 4x 2 u xy − 2x 2 u xytt + 5yu xyy − x 2 u xyyy + 4x yu xx + 2x yu xxtt + 10xu xxy + 4x yu xxyy − 7yu xxx + 2x 2 u xxxy − 3y 2 u xxxy − 2x yu xxxx .


457

S Jamal and A H Kara Q12 = yu xx + x 2 u yyy − yu yy + y 2 u yxx − 2x yu yyx − 2xu xy , 1 −u t −yu yy + x 2 u yyy − 2xu xy − 2x yu xyy + yu xx + y 2 u xxy t12 = 2 + u −yu yyt + x 2 u yyyt − 2xu xyt − 2x yu xyyt + yu xxt + y 2 u xxyt , 1 x 12 = − 2x yu 2y − 2x yu y u ytt − xu y u yy − 2x yu 2yy + 2x yu y u yyy + 2y 2 u y u x 6 + y 2 u ytt u x − 2yu yy u x + 3x 2 u yyy u x − y 2 u yyy u x + y 2 u y u xtt + 2y 2 u yy u xy − 6xu x u xy + u tt (xu y − y(−2xu yy + u x + 2yu xy ) − y 2 u y u xyy − 6x yu x u xyy − xu y u xx − 2x yu yy u xx + 4yu x u xx + 2y 2 u xy u xx + 2x yu y u xxy + 2y 2 u x u xxy − y 2 u y u xxx + u 2xu y + xu ytt + 4x yu yy + 2x yu yytt − 7xu yyy − 2x yu yyyy − 2yu x − yu xtt + 6u xy − 4y 2 u xy − 2y 2 u xytt + 10yu xyy − 3x 2 u xyyy + 2y 2 u xyyy + 5xu xxy + 4x yu xxyy − 2yu xxx − y 2 u xxxy ,

y

12 =

1 −2u 2 +3x 2 u 2y +3x 2 u y u ytt −4yu y u yy +3x 2 u 2yy −4x yu y u x − xu yy u x 6 + 2x yu yyy u x + y 2 u 2x − 2x yu y u xtt + y 2 u x u xtt − 6xu y u xy − 4x yu y u xyy − y 2 u x u xyy + 2yu y u xx + 3x 2 u yy u xx + y 2 u yy u xx − xu x u xx − 4x yu xy u xx + y 2 u 2xx + u tt yu y − 3x 2 u yy + xu x + 4x yu xy − y 2 u xx − 3x 2 u y u xxy + 3y 2 u y u xxy + 2x yu x u xxy + 2x yu y u xxx − y 2 u x u xxx + u − 2u tt + 2yu y + yu ytt + 5u yy − 6x 2 u yy − 3x 2 u yytt + 2yu yyy + 2xu x + xu xtt + 8x yu xy + 4x yu xytt + 11xu xyy + 2x yu xyyy − u xx − 2y 2 u xx − y 2 u xxtt − 10yu xxy + 3x 2 u xxyy − 2y 2 u xxyy −xu xxx − 4x yu xxxy + y 2 u xxxx − 4x yu yy u xy − 2x yu ytt u x .

1 1 Q13 = −x yu xx − x 3 u yyy + y 3 u xxx + x yu yy − y 2 u xy − x y 2 u xxy 3 3 + yx 2 u xyy + x 2 u xy , 1 u t − 3x yu yy + x 3 u yyy − 3x 2 u xy + 3y 2 u xy − 3x 2 yu xyy + 3x yu xx t13 = 6 + 3x y 2 u xxy − y 3 u xxx + u 3x yu yyt − x 3 u yyyt + 3x 2 u xyt − 3y 2 u xyt + 3x 2 yu xyyt − 3x yu xxt − 3x y 2 u xxyt + y 3 u xxxt , 1 x − 4yu 2 + 2x 2 yu 2y + 2x 2 yu y u ytt + x 2 u y u yy − y 2 u y u yy + 2x 2 yu 2yy 13 = 12 − 2x 2 yu y u yyy − 4x y 2 u y u x − 2x y 2 u ytt u x + 4x yu yy u x − 2x 3 u yyy u x + 2x y 2 u yyy u x + 2y 3 u 2x − 2x y 2 u y u xtt + 2y 3 u x u xtt − 4x y 2 u yy u xy − 6y 2 u x u xy + 2x y 2 u y u xyy + 6x 2 yu x u xyy − 2y 3 u x u xyy + x 2 u y u xx − y 2 u y u xx + 2x 2 yu yy u xx + 2y 3 u yy u xx − 8x yu x u xx − 4x y 2 u xy u xx +2y 3 u 2xx +u tt − x 2 + y 2 u y −2y x 2 u yy − xu x −2x yu xy + y 2 u xx − 2x 2 yu y u xxy − 4x y 2 u x u xxy + 2x y 2 u y u xxx + u − 4yu tt − 2 x 2 − y 2 u y − x 2 u ytt + y 2 u ytt − 2yu yy − 4x 2 yu yy 458


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

y

13

− 2x 2 yu yytt + 7x 2 u yyy − y 2 u yyy + 2x 2 yu yyyy + 4x yu x + 2x yu xtt − 12xu xy + 8x y 2 u xy + 4x y 2 u xytt + 2x 3 u xyyy − 4x y 2 u xyyy + 10yu xx − 4y 3 u xx − 2y 3 u xxtt − 5x 2 u xxy − 4x 2 yu xxyy + 2y 3 u xxyy + 4x yu xxx + 2x y 2 u xxxy − 20x yu xyy + 11y 2 u xxy + 6x 2 u x u xy , 1 4xu 2 − 2x 3 u 2y − 2x 3 u y u ytt + 8x yu y u yy − 2x 3 u 2yy + 4x 2 yu y u x = 12 + 2x 2 yu ytt u x + x 2 u yy u x − y 2 u yy u x − 2x 2 yu yyy u x − 2x y 2 u 2x + 2x 2 yu y u xtt − 2x y 2 u x u xtt + 6x 2 u y u xy − 6y 2 u y u xy + 4x 2 yu yy u xy + 4x 2 yu y u xyy + 2x y 2 u x u xyy − 4x yu y u xx − 2x 3 u yy u xx − 2x y 2 u yy u xx + x 2 u x u xx − y 2 u x u xx + 4x 2 yu xy u xx − 2x y 2 u 2xx + u tt −2x yu y + 2x 3 u yy − x 2 u x + y 2 u x − 4x 2 yu xy + 2x y 2 u xx + 2x 3 u y u xxy − 6x y 2 u y u xxy − 2x 2 yu x u xxy − 2x 2 yu y u xxx + 2y 3 u y u xxx + 2x y 2 u x u xxx + u 4xu tt − 4x yu y − 2x yu ytt − 10xu yy + 4x 3 u yy + 2x 3 u yytt − 4x yu yyy − 2x 2 u x + 2y 2 u x − x 2 u xtt + 12yu xy − 8x 2 yu xy − 4x 2 yu xytt − 11x 2 u xyy + 5y 2 u xyy − 2x 2 yu xyyy + 4x y 2 u xx + 2x y 2 u xxtt + 20x yu xxy − 2x 3 u xxyy + 4x y 2 u xxyy + x 2 u xxx − 7y 2 u xxx + 4x 2 yu xxxy − 2y 3 u xxxy − 2x y 2 u xxxx + 2xu xx +y 2 u xtt .

We now select a few of the symmetries above and prove that they are variational using Theorem 1 where X = Qi ∂u . X E + AF Q4 E = Q x x + Q yy − Q tt − Q + (−Dxxx ) (u xx + u yy − u tt − u) = u xxxxx + u xxxyy − u xxxtt − u xxx − u xxxxx − u xxxyy + u xxxtt + u xxx = 0, X E + AF Q8 E = Q x x + Q yy − Q tt − Q + Dxy + y Dxyy − x Dyyy (u xx + u yy − u tt − u) = u xxxy + u xyyy − u xytt − u xy + y(u xxxyy + u xyyyy − u ttxyy − u xyy ) − x(u xxyyy + u yyyyy − u ttyyy − u yyy ) + 2u xyyy + xu xxyyy − yu xxxyy − u xxxy + u yyyyy − 2u xyyy − yu xyyyy − u xyyy − xu yyytt + yu xyytt + u xytt − xu yyy + yu xyy + u xy = 0, X E + AF Q11 E = Q x x + Q yy − Q tt − Q + (x Dxx + 2y Dxy − x Dyy − y 2 Dxxx + 2x y Dxxy − x 2 Dxyy )(u xx + u yy − u tt − u) = x (u xxxx + u xxyy − u xxtt − u xx ) + 2y(u xxxy + u xyyy − u ttxyy − u xy ) − x(u xxyy + u yyyyy − u ttyy − u yy ) − y 2 (u xxxxx + u xxxyy − u xxxtt − u xxx ) Pramana – J. Phys., Vol. 77, No. 3, September 2011

459

S Jamal and A H Kara + 2x y(u xxxxy + u xxyyy − u ttxxy − u xxy ) − x 2 (u xxxyy + u yyyyx − u xyytt − u xyy ) + 4u xyy + 4xu xxyy + x 2 u xxxyy − 4yu xxxyy − 2x yu xxxxy + xu xxyy − xu xxxx + y 2 u xxxxx − 2yu xxxy + x 2 u xyyyy − 4xu xxyy − 2x yu xxyyy − xu xxyy + 2yu xxxy + 2u xxx + 2yu xxxy + y 2 u xxxyy − 4u xyy − 2yu xyyy − x 2 u xyytt + 2x yu xxytt − xu yytt + xu xxtt − y 2 u xxxtt + 2yu xytt − 2u xxx − x 2 u xyy + 2x yu xxy − xu yy + xu xx − y 2 u xxx + 2yu xy + xu yyyy = 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 determine possible higher-order symmetries. These were obtained by investigating the classical (1+1) Gordon-type equations and, more formally, using the multiplier approach. It turned out 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 also calculated. 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)

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