A symmetry classification for a class of (2+1)-nonlinear wave equation
Mehdi Nadjafikhah, Rohollah Bakhshandeh-Chamazkoti, Ali Mahdipour-Shirayeh
aa r X i v : . [ m a t h . DG ] A ug A symmetry classification for a class of(2 + 1)-nonlinear wave equation
M. Nadjafikhah ∗ R. Bakhshandeh Chamazkoti † A. Mahdipour-Shirayeh ‡ Abstract
In this paper, a symmetry classification of a (2 + 1)-nonlinear waveequation u tt − f ( u )( u xx + u yy ) = 0 where f ( u ) is a smooth function on u , using Lie group method, is given. The basic infinitesimal methodfor calculating symmetry groups is presented, and used to determinethe general symmetry group of this (2 + 1)-nonlinear wave equation. It is well known that the symmetry group method plays an important rolein the analysis of differential equations. The history of group classificationmethods goes back to Sophus Lie. The first paper on this subject is [1],where Lie proves that a linear two-dimensional second-order PDE may admitat most a three-parameter invariance group (apart from the trivial infiniteparameter symmetry group, which is due to linearity). He computed themaximal invariance group of the one-dimensional heat conductivity equationand utilized this symmetry to construct its explicit solutions. Saying it themodern way, he performed symmetry reduction of the heat equation. Nowa-days symmetry reduction is one of the most powerful tools for solving nonlin-ear partial differential equations (PDEs). Recently, there have been several ∗ e-mail: m nadjafi[email protected] Department of Mathematics, Iran University of Sci-ence and Technology, Narmak, Tehran, IRAN. † e-mail: r [email protected] ‡ e-mail: [email protected] u tt = [ f ( u ) u x ] x , (1)in [3] and later its the generalized cases u tt = [ f ( x, u ) u x ] x , u tt = [ f ( u ) u x + g ( x, u )] x , (2)in [4] and [5], respectively, are investigated. Also the most important classesof the (1 + 1)-nonlinear wave equations with the forms v tt = f ( x, v x ) v xx + g ( x, v x ) , u tt = f ( x, u ) u xx + g ( x, u ) , (3)can be found in two attempts [6] and [7] respectively. An alternative formof Eq. (1) was also investigated by Oron and Rosenau [8] and Suhubi andBakkaloglu [9]. The equations u tt = F ( u ) u xx , u tt + K ( u ) u t = F ( u ) u xx , u tt + K ( u ) u t = F ( u ) u xx + H ( u ) u x , (4)are classified in [10, 11, 12], respectively. Lahno et al. [13] presented themost extensive list of symmetries of the equations u tt = u xx + F ( t, x, u, u x ) , (5)by using the infinitesimal Lie method, the technique of equivalence trans-formations, and the theory of classification of abstract low-dimensional Liealgebras. There are also some papers [14, 15, 16] devoted to the group clas-sification of the equations of the following form: u tt = F ( u xx ) , u tt = F ( u x ) u xx + H ( u x ) , u tt + u xx = g ( u, u x ) , (6)2tudies have also been made for (2+1)-nonlinear wave equation with constantcoefficients [17, 18, 19]. In the special case the (2 + 1)-dimensional nonlinearwave equation u tt = u n ( u xx + u yy ) , (7)is investigated in [20]. The goal of this paper is to investigate the Lie sym-metries for some class of (2 + 1)-nonlinear wave equation u tt − f ( u )( u xx + u yy ) = 0 , (8)where f ( u ) is a arbitrary smooth function of the variable u . Clearly, in Eq.(8) case of f u = 0 namely f ( u ) = constant is not interest because this casereduces the wave equation to a linear one. Similarly technics were applicablefor some classes of the nonlinear heat equations in [21, 22]. Let a partial differential equation contains one dependent variable and p independent variables. The one-parameter Lie group of transformations x i = x i + ǫξ i ( x, u ) + O ( ǫ ); u = u + ǫϕ ( x, u ) + O ( ǫ ) , (9)where i = 1 , . . . , p , and ξ i = ∂x i ∂ǫ | ǫ =0 , acting on ( x, u )-space has as its infinites-imal generator v = ξ i ∂∂x i + ϕ ∂∂u , i = 1 , . . . , p . (10)Therefore, the characteristic of the vector field v given by (10) is the function Q ( x, u (1) ) = ϕ ( x, u ) − p X i =1 ξ i ( x, u ) ∂u∂x i . (11)The symmetry generator associated with (10) given by v = ξ ∂∂x + η ∂∂y + τ ∂∂t + ϕ ∂∂u . (12)3he second prolongation of v is the vector field v (2) = v + ϕ x ∂∂u x + ϕ y ∂∂u y + ϕ t ∂∂u t + ϕ xx ∂∂u xx + ϕ xy ∂∂u xy + ϕ xt ∂∂u xt (13)+ ϕ yy ∂∂u yy + ϕ yt ∂∂u yt + ϕ tt ∂∂u tt . with coefficient ϕ ι = D ι Q + ξu xι + ηu yι + τ u tι , (14) ϕ ι = D ι ( D Q ) + ξu xι + ηu yι + τ u tι , (15)where Q = ϕ − ξu x − ηu y − τ u t is the characteristic of the vector field v givenby (12) and D i represents total derivative and subscripts of u are derivativewith respect to the respective coordinates. ι and in above could be x, y or t coordinates. By the theorem 6.5. in [23], v (2) [ u tt − f ( u )( u xx + u yy )] = 0whenever u tt − f ( u )( u xx + u yy ) = 0 . (16)Since v (2) [ u tt − f ( u )( u xx + u yy )] = ϕ tt − ϕf u ( u xx + u yy ) − f ( u )( ϕ xx + ϕ yy ) , therefore ϕ tt − ϕf u ( u xx + u yy ) − f ( u )( ϕ xx + ϕ yy ) = 0 . (17)Using the formula (15) we obtain coefficient functions ϕ xx , ϕ yy , ϕ tt as ϕ xx = D x Q + ξu xx + ηu yxx + τ u txx , (18) ϕ yy = D y Q + ξu xyy + ηu yyy + τ u tyy , (19) ϕ tt = D t Q + ξu xtt + ηu ytt + τ u ttt , (20)where the operators D x , D y and D t denote the total derivatives with respectto x, y and t : D x = ∂∂x + u x ∂∂u + u xx ∂∂u x + u xy ∂∂u y + u xt ∂∂u t + . . .D y = ∂∂y + u y ∂∂u + u yy ∂∂u y + u yx ∂∂u x + u yt ∂∂u t + . . . (21) D t = ∂∂t + u t ∂∂u + u tt ∂∂u t + u tx ∂∂u x + u ty ∂∂u y + . . . u ( x, y, t ) whosecoefficients are certain derivatives of ξ, η, τ and ϕ . Since ξ, η, τ, ϕ only dependon x, y, t, u we can equate the individual coefficients to zero, leading to thecomplete set of determining equations: ξ = ξ ( x, t ) (22) η = η ( y, t ) (23) τ = τ ( x, y, t ) (24) ϕ = α ( x, y, t ) u + β ( x, y, t ) (25) τ t = ϕ u = α ( x, y, t ) , (26) ξ tt = f ( u )( ξ xx − ϕ xu ) (27) η tt = f ( u )( η yy − ϕ yu ) (28) τ tt = f ( u )( τ xx + τ yy ) + 2 ϕ tu (29) f u ϕ = 2 f ( u )( ξ x − τ t ) (30) f u ϕ = 2 f ( u )( η y − τ t ) (31) f ( u ) τ x = ξ t (32) f ( u ) τ y = η t (33) ϕ tt = f ( u )( ϕ xx + ϕ yy ) (34) In this section we start to classify the symmetries of the nonlinear waveequation (8). To fined a complete solution of the above system we considerEq. (30) and with assumption f u = 0 we rewrite: ϕ = 2 ff u ( ξ x − τ t ) (35)Note the case of f ( u ) = constant explained in introduction. Two generalcases are possible: i) ff u = c, (36)ii) ff u = g ( u ) , (37)5here c is a constant. (i) In this case with integrating from Eq. (36) with respect to u to obtain f ( u ) = Ke uc , (38)where K is an integration constant. Then the Eq. (35) reduce to ϕ = 2 c ( ξ x − τ t ) . (39)With substituting (39) into (26)-(33) we have ξ ( x ) = c x + c ; η ( y ) = c y + c ; (40) τ ( t ) = c t + c ; ϕ = 2 c ( c − c ) . where c i , i = 1 , . . . ,
5, are arbitrary constants. The Lie symmetry generatorfor Eq. (8) in this case (i) is v = ( c x + c ) ∂∂x + ( c y + c ) ∂∂y + ( c t + c ) ∂∂t + 2 c ( c − c ) ∂∂u . (41)Therefore the symmetry algebra of the (2 + 1)-nonlinear wave equation (8)is spanned by the vector fields v = x∂ x + y∂ y + 2 c∂ u ; v = ∂ x ; v = ∂ y ; v = t∂ t − c∂ u ; v = ∂ t . (42)The commutation relations satisfied by generators (42) in the case (i) areshown in table 1. The invariants associated with the infinitesimal generator v are obtained by integrating the characteristic equation: dxx = dyy = dt du c . (43)and have the forms r = yx , s = t, and ω ( r, s ) = u ( x, y, t ) − c ln x, (44)6able 1: Commutation relations satisfied by infinisimal generators in Cases(i) and (ii)[ v i , v j ] v v v v v v − v − v v v v v v − v v v ω toobtain ω ss = Ke ωc (cid:16) (1 + r ) ω rr + 2 rω r − c (cid:17) , (45)By solving this partional differential equation we obtain the reduced equation ω ( r, s ) = ζ ( r ) + ζ ( s ) , (46)where ζ and ζ satisfy in following second-order differential equations¨ ζ ( r + 1) + c e − ζ c + 2( r ˙ ζ − c ) = 0; ¨ ζ + Kc e ζ c = 0 , (47)with c , c, K, arbitrary constants. The characteristic equation associated with v is dx dy dtt = du − c , (48)which generate the invariants x , y , t − c e − u . Then the similarity solution ischosen to have the form u ( x, y, t ) = 2 c ln h ( x, y ) t . (49)By substituting (49) into (16) to determine the form of the function h toobtain 1 K − h ( x, y )( h xx + h yy ) + h x + h y = 0 , (50)7hich has the solution h ( x, y ) = mx + py + q ; m + p = K − , (51)where m, p, q are arbitrary constants. For the remaining infinitesimalgenerators v , v , v , the invariants associated are the arbitrary functions λ ( y, t, u ), µ ( x, t, u ), and ν ( x, y, u ) respectively. (ii) In this case we classify solution of the wave equation (8), with assumption g u = 0. With substituting (34) into (27)-(28), since ξ , η and τ are notdependent to u , therefore from u ( x, y, t, u ) = 2 g ( u )( ξ x − τ t ) , (52)and also from (8) and (25), we conclude g ( u ) = e u + e , (53)where e = 0 and e are arbitrary constants. Now we substitute (53) into(37) and rewrite f u f = 1 e u + e . (54)Therefore by integrating from (54) with respect to u we have f ( u ) = L ( e u + e ) e , (55)where L is an integration constant. Now by considering Eq. (22)-(34), it isnot hard to find that the components ξ , η , τ and ϕ of infinitesimal generatorsbecome ξ ( x ) = c x + c ; η ( y ) = c y + c ; (56) τ ( t ) = c t + c ; ϕ = 2 e ( c − c ) u + 2 e ( c − c ) , where c i , i = 1 , . . . ,
5, are arbitrary constants. From above the five infinites-imal generators can be constructed: v = x∂ x + y∂ y + (2 e u + 2 e ) ∂ u ; v = ∂ x ; v = ∂ y ; v = t∂ t − e u + e ) ∂ u ; v = ∂ t . (57)8t is easy to check that the infinitesimal generators (57) from a closed Liealgebra whose it’s corresponding commutation relations are coincided withobtained results in table 1. For generator v , the associated equations are dxx = dyy = dt du e u + e ) , (58)which generate the invariants p = yx , q = t , and ϑ ( p, q ) = ( u + e e ) x − e .Consequently, the similarity solution is chosen to have the form u ( x, y, t ) = ϑ ( t, yx ) x e + e e . (59)We substitute (60) into (16) to obtain following partional differential equation ϑ pp = Lϑ e [( q + 1) ϑ qq + 2 qϑ q (1 − e ) + 2 e (2 e − ϑ ] , (60)as an example, for particular case e = 1, the solution of (60) is ϑ ( p, q ) = ς ( p ) · ς ( q ) , (61)where ς ( p ) and ς ( q ) satisfy in second order equations¨ ς − cς = 0; ( q + 1) ¨ ς − q ˙ ς + 2 ς − cL − = 0 , (62)where c is a arbitrary constant. Also characteristic equation correspondinggenerator v is dx dy dtt = du − e u + e ) , (63)and so u ( x, y, t ) = l ( x, y ) t − e − e e . (64)Substitute (64) into (16), l ( x, y ) satisfies in the following equation: L ( l xx + l yy ) l ( e − − − e (2 e + 1) = 0 . (65)For the remaining infinitesimal generators v , v , v , the invariants associ-ated are the arbitrary functions r ( y, t, u ), m ( x, t, u ), and n ( x, y, u ) respec-tively. 9 Conclusion and new ideas
In this paper we have obtained some particular Lie point symmetries groupof the (2 + 1)-nonlinear wave equation u tt − f ( u )( u xx + u yy ) = 0 where f ( u )is a smooth function on u , by using here the classical Lie symmetric method.In section 2, the complete set of determining equations was obtained bysubstituting the equations (18), (19) and (20) in invariance condition (17) andthen in section 3, we classify the symmetries of this nonlinear wave equationby assumption two cases in (36) and (37) to consider ff u is a constant or isa smooth function with respect to u and f u = 0. The commutation relationssatisfied by infinitesimal generators in two cases are given in table 1, andtheir invariants associated with the infinitesimal generators are obtained.This method is suitable for preliminary group classification of some class ofnonlinear wave equations [6, 7].There are some classes of (2 + 1)-nonlinear wave equations that will beinvestigated by both classical or nonclassical symmetries method similarlywhose we do for classical case. For examples u tt − f ( x, u )( u xx + u yy ) = 0 , (66) u tt − f ( x, u x )( u xx + u yy ) = 0 , (67)or generalized case u tt − f ( x, y, u, u x )( u xx + u yy ) = 0 , (68)are interested. It is a pleasure to thank the anonymous referees for their constructive sugges-tions and helpful comments which have materially improved the presentationof the paper.
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