CHINESE JOURNAL OF PHYSICS VOL. 51, NO. 6 December 01 Symmery Group Theorem of he Lin-Tsien Equaion and Conservaion Laws Relaing o he Symmery of he Equaion Xi-Zhong Liu and Jun Yu Insiue of Nonlinear Science, Shaoxing Universiy, Shaoxing 1000, China (Received November 1, 01; Revised March 6, 01) We derive he symmery group heorem for he Lin-Tsien equaion by using he modified CK direc mehod, from which we obain he corresponding symmery group. Conservaion laws corresponding o he Kac-Moody-Virasoro symmery algebra of he Lin-Tsien equaion are obained up o second order group invarians. DOI: 10.61/CJP.51.111 PACS numbers: 0.0.Jr, 47.10.ab, 0.0.Ik I. INTRODUCTION Conservaion laws which originae in mechanics and physics play an imporan role in physics and mahemaics. Nonlinear parial differenial equaions (NPDEs) ha admi conservaion laws arise in many disciplines of he applied sciences including physical chemisry, fluid mechanics, paricle and quanum physics, plasma physics, elasiciy, gas dynamics, elecromagneism, magneohydro-dynamics, nonlinear opics, and he bio-sciences. Especially in solion heory conservaion laws have many significan uses, paricularly wih regard o inegrabiliy and linearizaion, he analysis of soluions, and numerical soluion mehods. Furhermore, compleely inegrable NPDEs [1, ] admi infiniely many independen conservaion laws. Besides, finding he symmery of NPDEs is also very imporan (see, e.g., Refs. [ 6]). The mahemaical foundaions for he deerminaion of he full group for a sysem of differenial equaions can be found in Ames [7] and Bluman and Cole [8], and he general heory is found in Ovsiannikov [9]. Among hem, he modified CK direc mehod [10, 11] is an effecive mehod for finding symmeries [1, 1], and one advanage of his mehod is ha one can easily obain he relaionship beween new exac soluions and old ones of he given NPDEs. A conservaion law is closely conneced wih a symmery, and his connecion is given by he famous Noeher heorem. In he classical Noeher heorem [14], if a given sysem of differenial equaions has a variaional principle, hen a coninuous symmery ha leaves invarian he acion funcional o wihin a divergence yields a conservaion law [15 18]. The Noeher heorem has been he only general device allowing one, in he class of Euler- Lagrange equaions, o reduce he search for conservaion laws o a search for symmeries. In he las few years, effecive mehods have been devised for finding conservaion laws for he very special class of so-called Lax equaions. In 000, Kara [19] presened he Elecronic address: liuxizhong1@16.com hp://psroc.phys.nu.edu.w/cjp 111 c 01 THE PHYSICAL SOCIETY OF THE REPUBLIC OF CHINA
11 SYMMETRY GROUP THEOREM TO THE LIN-TSIEN... VOL. 51 direc relaionship beween he conserved vecor of a PDE and he Lie-Bäcklund symmery generaors of he PDE, from which i is possible for us o obain conservaion laws from symmeries (see, e.g. [0]). The Lin-Tsien equaion [1] u x + u x u xx u yy = 0, (1) where he variable u u(x, y, ) is a velociy poenial, is known as a compleely inegrable model. I has been widely used o sudy dynamic ransonic flow in wo space dimensions, plasma physics, opics, condensed maer physics, ec. In he presen reamen of he Lin-Tsien equaion, our principal aims are o examine is symmery groups horoughly and hen o find he implicaions of hese symmeries as regards o conservaion laws. This paper is organized as follows. In Secion II, we derive he symmery group heorem for he Lin-Tsien equaion by using he modified CK direc mehod, hen he corresponding Lie poin symmery groups and infinie dimensional Kac-Moody-Virasoro (KMV) symmery algebra [] are obained sraighforwardly. As a comparison, we also derive he Lie poin symmery groups by he radiional Lie approaches and he resul shows ha boh mehods produce he same resuls. In Secion III, we firs review some basic noions abou Lie-Bäcklund operaors and hen using his we finally obain conservaion laws of he infinie dimensional Kac-Moody-Virasoro symmery algebra ha he Lin-Tsien equaion possesses up o second order group invarians. I is emphasized ha equaions wih he same symmeries may possesses he same ypes of conservaion laws. The las secion is a shor summary and discussion. II. TRANSFORMATION GROUP BY THE DIRECT METHOD AND KAC- MOODY VIRASORO STRUCTURE OF THE LIE POINT SYMMETRY ALGEBRA To find he complee poin symmery ransformaion group of (1), one should find he general ransformaions in he following form u = U(x, y,, F (ξ, η, τ)), () where ξ, η, and τ are funcions of x, y, and should be deermined by requiring ha F (ξ, η, τ) saisfies he same (+1)-dimensional equaion as u = u(x, y, ) wih he ransformaion {u, x, y, } {F, ξ, η, τ}, i.e., F τ,ξ + F ξ F ξξ F ηη = 0. () Forunaely, we can prove ha for he Lin-Tsien equaion i is enough o ake u = α + βf (ξ, η, τ), (4) insead of (), where α, β, ξ, η, and τ are funcions of {x, y, }. To prove he conclusion (4), one should submi he general expression () o Eq. (1). Afer eliminaing F ηη and heir higher derivaives via () and vanishing all he coefficiens
VOL. 51 XI-ZHONG LIU AND JUN YU 11 of he differen erms of he derivaives of he funcions F, one can ge many complicaed deermining equaions for he 4 funcions U U(x, y,, F (ξ, η, τ)), ξ, η, and τ. Two of hem read as ξ x η xu F F ξ = 0, τ x ξ xu F F τ = 0. For he reason ha U F should no be zero, and here is no nonrivial soluion for ξ x = 0, he only way o cause he coefficiens of F ξ and F τ o vanish is η x = 0, τ x = 0. (5) Under he condiion (5), he deermining equaion conaining F ξ is (U F ξ x η y)f ξ = 0. Solving he above equaion for U hen proves he conclusion ha assumpion (4) insead of he general one () is sufficien o find he general symmery group of he Lin-Tsien equaion. Now he subsiuion of (4) wih (5) ino he Lin-Tsien equaion leads o (β y η y + β x η βη yy )F η + ξ x β(ξ xx β + ξ x β x )F ξ + (β(ξ xβ + τ y ξ y τ ξ x )F ξξ +(β x βξ xx + β xξ x + βξ x β xx )F + βξ x + α x βξ xx + α x β x ξ x β y ξ y + βξ x α xx +β ξ x + β x ξ βξ yy )F ξ βτ y F ττ + (β x τ β y τ y βτ yy )F τ +(β x βξ xf + β(α x ξ x + ξ ξ x ξ y))f ξξ + β(η y + τ ξ x τ y ξ y )F ηη +β(η ξ x η y ξ y )F ξη + β x β xx F + (β x + β x α xx + α x β xx β yy )F +α x + α x α xx α yy βη y τ y F τη = 0. (6) Eq. (6) is rue for arbirary soluions F only when all he coefficiens of he polynomials of he derivaives of F are zero, which leads o a sysem of deermining equaions for ξ, η, τ, α, and β β y η y + β x η βη yy = 0, ξ x β(ξ xx β + ξ x β x ) = 0, (7) βξ x + α x βξ xx + α x β x ξ x β y ξ y + βξ x α xx + β ξ x + β x ξ βξ yy = 0, (8) β x βξ xx + β xξ x + βξ x β xx = 0, βτ y = 0, β x τ β y τ y βτ yy = 0, (9) (β x + β x α xx + α x β xx β yy ) = 0, β(η y + τ ξ x τ y ξ y ) = 0, (10) β(ξ xβ + τ y ξ y τ ξ x ) = 0, β x βξ x = 0, β(η ξ x η y ξ y ) = 0, (11) β x β xx = 0, β(α x ξ x + ξ ξ x ξ y) = 0, α x + α x α xx α yy = 0, (1)
114 SYMMETRY GROUP THEOREM TO THE LIN-TSIEN... VOL. 51 βη y τ y = 0. (1) I is sraighforward o obain he general soluions of he deermining equaions (7) (1). The resuls are ξ = τ 1 x + 1 y τ + τ yη 0 τ 1 + ξ 0, η = τ y + η 0, β = τ 1, (14) α = (6τ τ τ + 8τ + 9τ τ )y 4 81τ (η 0τ τ + η 0 τ τ η 0 τ + τη 0 )y τ 8 + (4τ + τ τ )x 9τ 4τ ξ 0 τ 6η 0 η 0 τ + 6ξ 0 τ + 5τ η0 y τ 7 + (η 0τ + η 0 τ )x + α τ 5 y τ x (ξ 0τ η0 )x + α τ τ 4 1, (15) where ξ 0 ξ 0 (), η 0 η 0 (), τ τ(), α 1 α 1 (), and α α () are arbirary funcions of ime. In summary, he following heorem holds: Theorem 1: If F = F (x, y, ) is a soluion of he Lin-Tsien equaion (1), hen so is u = (6τ τ τ + 8τ + 9τ τ )y 4 81τ (η 0τ τ + η 0 τ τ η 0 τ + τη 0 )y τ 8 + (4τ + τ τ )x 9τ 4τ ξ 0 τ 6η 0 η 0 τ + 6ξ 0 τ + 5τ η0 y τ 7 + (η 0τ + η 0 τ )x + α τ 5 y τ x (ξ 0τ η0 )x τ τ 4 +α 1 + τ 1 F (ξ, η, τ), (16) wih (14), where ξ 0, η 0, τ, α 1, and α are arbirary funcions of. Applying he heorem o some simple exac soluions wihou arbirary funcions, one may obain some ypes of novel generalized soluions wih some arbirary funcions. In he following, we jus presen one special soluion example.
VOL. 51 XI-ZHONG LIU AND JUN YU 115 Example 1. I is quie rivial ha he Lin-Tsien equaion (1) possesses a special simple soluion F = 1. (17) Using he ransformaion heorem on he above special soluion we have he following new special soluion of he Lin-Tsien equaion: u = (6τ τ τ + 8τ + 9τ τ )y 4 81τ (η 0τ τ + η 0 τ τ η 0 τ + τη 0 )y τ 8 + (4τ + τ τ )x 9τ 4τ ξ 0 τ 6η 0 η 0 τ + 6ξ 0 τ + 5τ η0 y τ 7 + (η 0τ + η 0 τ )x + α τ 5 y τ x (ξ 0τ η0 )x 1 + α τ τ 4 1 + τ. (18) In he radiional Lie group heory, one always ries o find he Lie poin symmeries firs and hen use Lie s firs fundamenal heorem o obain he symmery ransformaion group. Conversely, we are forunae o obain he symmery ransformaion group in he firs place by a simple direc mehod. Once he ransformaion group is known, he Lie poin symmeries and he relaed Lie symmery algebra can be obained sraighforwardly by a more simple limiing procedure. For he Lin-Tsien (1), he corresponding Lie poin symmeries can be derived from he symmery group ransformaion heorem by seing η 0 () = ϵh(), τ() = + ϵf(), ξ 0 () = ϵg(), α 1 () = ϵm(), α () = ϵn(), (19) wih ϵ being an infiniesimal parameer, hen (16) can be wrien as u = F + ϵσ(f ) + O(ϵ ), ( σ(f ) = g() + 1 f ()x + 1 ) ( f ()y + yh () F x + h() + ) f ()y F y +ff + 1 f ()F f xy 1 f x g ()x + m() +n()y 1 9 f y 4 h ()y g y h xy. (0)
116 SYMMETRY GROUP THEOREM TO THE LIN-TSIEN... VOL. 51 The equivalen vecor expression of he above symmery reads { V = f ()x + 1 ) f ()y x + f ()y y + f() f ()F f xy 1 f x 1 ) } 9 f y 4 F { + g() x + (g ()x + g y ) } { + yh () F x + h() y ( ) } { + h ()y + h xy m() } { n()y } F F F V 1 (f()) + V (g()) + V (h()) + V 4 (m()) + V 5 (n()). (1) Since he funcions f, g, h, m, and n are arbirary, he corresponding Lie algebra is an infinie dimensional Lie algebra. I is easy o verify ha he symmeries V i, i = 1,,, 4, 5 consiue an infinie dimensional Kac-Moody-Virasoro [] ype symmery algebra S wih he following nonzero commuaion relaions: [V 1 (f), V 4 (m)] = V 4 mf + fm ), () [V (g), V (h)] = V 5 ((hg gh + h g )), () [V 1 (f), V (h)] = V (fh ) hf, (4) [V 1 (f), V (g)] = V (g f 1 ) gf ), (5) [V (g 1 ), V (g )] = V 4 (g 1 g g 1 g ), (6) [V (h), V 5 (n)] = V 4 (hn), (7) [V 1 (f), V 5 (n)] = V 5 (f n + fn ), (8) [V 1 (f 1 ), V 1 (f )] = V 1 (f 1 f f f 1 ), (9) [V (h 1 ), V (h )] = V (h 1 h h h 1 ). (0)
VOL. 51 XI-ZHONG LIU AND JUN YU 117 I should be emphasized ha he algebra is infinie dimensional, because he generaors V 1, V, V, V 4, and V 5 all conain arbirary funcions. The algebra is closed because all he commuaors can be expressed by he generaors belonging o he generaor se, usually wih differen funcions, and he generaors conaining differen funcions belong o he se. Especially, i is clear ha he symmery V 1 (f) consiue a cenerless Virasoro symmery algebra. As a comparison, we now derive he Lie poin symmery of he Lin-Tsien equaion by he sandard Lie approach briefly. To sudy he symmery of Equaion (1), we search for he Lie poin symmery ransformaions in he vecor form V = X x + Y y + T + U u, where X, Y, T, and U are funcions wih respec o x, y,, u, which means ha (1) is invarian under he poin ransformaion {x, y,, u} {x + ϵx, y + ϵy, + ϵt, u + ϵu} wih infiniesimal parameer ϵ. In oher words, he symmery of he equaion (1) can be wrien as he funcion form σ = Xu x + Y u y + T u U, (1) where he symmery σ is a soluion of he linearized equaion for (1) σ x + σ x u xx + u x σ xx σ yy = 0, () which is obained by subsiuing u = u + ϵσ ino (1) and dropping he nonlinear erms in σ. I is easy o solve ou X(x, y,, u), Y (x, y,, u), T (x, y,, u), and U(x, y,, u) by subsiuing (1) ino (), and eliminaing u yy and is higher order derivaives by means of he Lin-Tsien equaion. Afer aking he consans as zero, we ge he resuls X(x, y,, u) = 1 T x + 1 T y + X y + Y, () Y (x, y,, u) = T y + X, (4) T (x, y,, u) = T (), (5) U(x, y,, u) = 1 x T 1 ut + xy T + xx y + xy + 1 9 y4 T + X y + Y y + Z 1 y + Z, (6)
118 SYMMETRY GROUP THEOREM TO THE LIN-TSIEN... VOL. 51 where X, Y, T, Z 1, and Z are arbirary funcions of. The vecor form of he Lie poin symmeries reads V = T + 1 ) ( ) T y + X y + Y x + T + X y + T + ut 1 x T xy T xx y xy 1 9 y4 T X y Y y Z 1 y Z ) u, (7) which is exacly he same as ha obained by he modified CK approach. III. CONSERVATION LAWS RELATED TO THE THE SYMMETRY (7) In order o obain conservaion laws relaed o he symmery (7), we need some basic noions abou Lie-Bäcklund operaors firs. A Lie-Bäcklund operaor is given by X 0 = ξ i x i + η u + ζ i + ζ i1 i u +, (8) i u i1 i where ξ i, η, and he addiional coefficiens are ζ i = D i (W ) + ξ j u ij, ζ i1 i = D i1 i (W ) + ξ j u ji1 i, (9) and W is he Lie characerisic funcion defined by W = η ξ j u j (40) wih D i being he operaor of oal differeniaion as D i = x i + u i u + u ij +, i = 1,, n, (41) u j u i = D i (u), u ij = D j D i (u). (4) These definiions and resuls relaing o Lie-Bäcklund operaor can be found in [], and he repeaed indices mean summaions, which is known as he Einsein summaion rule. Using Equaions (8) (4), we can calculae he nd-order Lie-Bäcklund operaor of he vecor field V defined by Equaion (7): ξ x = 1 xt + 1 T y + X y + Y, (4)
VOL. 51 XI-ZHONG LIU AND JUN YU 119 ξ y = yt + X, (44) ξ = T, (45) η = 1 ut + 1 x T + xy T + xx y + xy + 1 9 y4 T + X y + Y y + Z 1 y + Z, (46) ζ x = T u x + xt + y T + X y + Y, (47) ζ y = T u y + 4 T yx + xx + 4 9 y T + X y + 4Y y ( ) +Z 1 T y + X u x, (48) ζ = 4 T u 1 ut + 1 x T + xy T + xyx + xy + 1 9 y4 T + X y + Y y + Z 1, y + Z, xt + 1 ) ( ) y T + yx + Y u x yt + X u y, (49) ζ xx = u xx T + T, (50) ζ xy = 4 T u xy + 4 ( ) yt + X yt + X u xx, (51) ζ x = 5 T u x T u x + xt + y T + X y + Y xt + 1 ) ( ) y T + yx + Y u xx T y + X u xy, (5) ζ yy = 5 T u yy + 4 xt + 4 y T + 4X y + 4Y u xt ( ) yt + X u xy, (5)
1140 SYMMETRY GROUP THEOREM TO THE LIN-TSIEN... VOL. 51 ζ y = T u y T u y + 4 xyt + xx + 4 9 y T + X y ( ) ( ) +4yY + Z 1, yt + X u x yt + X u x xt + 1 ) ( ) y T + yx + Y u xy yt + X u yy, (54) ζ = 7 T u 5 T u 1 T u + 1 x T + xy T + xyx +xy + 1 9 y4 T + X y + Y y + Z 1, y + Z, xt + 1 ) y T + yx + Y u x xt + 1 y T ( ) ( ) +yx + Y )u x yt + X u y yt + X u y. (55) Correspondingly, he second order Lie-Bäcklund operaor is given by X 0 = ξ x x + ξy y + ξ + η u + ζ x + ζ y + ζ + ζ u u x u y u u xx +ζ xy + ζ x + ζ yy + ζ y + ζ. (56) u xy u x u yy u y u Theorem ([19], [4]): Suppose ha X 0 is a Lie-Bäcklund symmery of (1) such ha he conservaion vecor T = (T 1, T, T ) is invarian under X 0. Then X 0 (T i ) + T i D j ξ j j=1 T j D j (ξ i ) = 0, (i = 1,, ), (57) j=1 where D 1 = D x, D = D y, D = D, and ξ i are deermined by (56). A Lie-Bäcklund symmery X 0 is said o be associaed wih a conserved vecor T of (1) if X 0 and T saisfy relaions (57). Now we consruc he corresponding conservaion laws relaing o (7) in he form D x J 1 + D y J + D ρ = 0, (58) where T 1 = J 1, T = J, T = ρ wih J 1, J, and ρ being funcions of {x, y,, u, u x, u y,, u }. Of which ρ is called he conserved densiy, J 1 and J are he conserved currens associaed wih space dimensions x and y, respecively.
VOL. 51 XI-ZHONG LIU AND JUN YU 1141 In erms of T = (J 1, J, ρ), Eq. (57) is equivalen o he following hree equaions: xt + 1 ) ( ) J1 y T + yx + Y x + yt J1 + X y + T J ( 1 + 1 ut + 1 x T + xy T + xyx + xy + 1 9 y4 T + ) X y + Y y J1 + Z 1 y + Z u ( + T u x + xt + ) [ J1 y T + X y + Y + T u y + 4 u x xyt + xx ) + 4 9 y T + X y + 4Y y + Z 1 ( yt + X u x ] J1 u y + [ 4 T u 1 ut + 1 x T + xy T + xyx + xy + 1 9 y4 T + X y + Y y + Z 1, y + Z, xt + 1 ) ( ) ] ( y T + yx + Y u x yt J1 + X u y + T u xx + ) u T J1 u xx [ + 4 T u xy + 4 ( ) ] [ yt + X yt J1 + X u xx + 5 u xy T u x T u x + xt + y T + X y + Y xt + 1 ) y T + yx + Y u xx ( ) ] [ yt J1 + X u xy + 5 u x T u yy + 4 xt + 4 y T + 4X y + 4Y u xt ( ) ] [ yt J1 + X u xy + u y T u y T + 4 u yy xyt + xx + 4 9 y T ( ) ( ) +X y + 4Y y + Z 1, yt + X u x yt + X u x xt + 1 ) ( ) ] [ y T + X y + Y u xy yt J1 + X u yy + 7 u y u T 5 u T 1 ut + 1 x T + xy T + xx y + xy + 1 9 y4 T + X y + Y y +Z 1, y + Z, xt + 1 ) y T + X y + Y u x xt + 1 y T ( ) ( ) ] +X y + Y )u x yt + X u y yt J1 + X u y + 5 u T J 1 ( X + ) yt J xt + 1 y T + yx + Y )ρ = 0, (59)
114 SYMMETRY GROUP THEOREM TO THE LIN-TSIEN... VOL. 51 xt + 1 ) ( ) J y T + yx + Y x + yt J + X y + T J ( + 1 ut + 1 x T + xy T + xyx + xy + 1 9 y4 T + ) X y + Y y J + Z 1 y + Z u ( + T u x + xt + ) [ J y T + X y + Y + T u y + 4 u x xyt + xx + 4 ( ) ] 9 y T + X y + 4Y y + Z 1 yt J + X u x u y [ + 4 T u 1 ut + 1 x T + xy T + xyx + xy + 1 9 y4 T + X y + Y y + Z 1, y + Z, xt + 1 ) y T + yx + Y u x ( ) ] ( yt J + X u y + T u xx + ) u T J u xx [ + 4 T u xy + 4 ( ) ] yt + X yt J + X u xx u xy [ + 5 T u x T u x + xt + y T + X y + Y xt + 1 ) ( ) ] [ y T + yx + Y u xx yt J + X u xy + 5 u x T u yy + 4 xt + 4 y T + 4X y + 4Y ( ) ] u xt yt J + X u xy u yy [ + u y T u y T + 4 xyt + xx + 4 9 y T + X y + 4Y y + Z 1, ( ) ( ) ( 1 yt + X u x yt + X u x xt + 1 ) y T + X y + Y u xy ( ) ] [ yt J + X u yy + 7 u y u T 5 u T 1 ut + 1 x T + xy T +xx y + xy + 1 9 y4 T + X y + Y y + Z 1, y + Z, xt + 1 ) y T + X y + Y u x xt + 1 ) y T + X y + Y u x ( ) ( ) ] yt + X u y yt J + X u y + 4 ( ) u T J yt + X ρ = 0, (60)
VOL. 51 XI-ZHONG LIU AND JUN YU 114 xt + 1 ) ( ) ρ ρ y T + yx + Y x + yt + X y + T ρ ( + 1 ut + 1 x T + xy T + xyx + xy + 1 9 y4 T + ) ρ X y + Y y + Z 1 y + Z u ( + T u x + xt + ) [ ρ y T + X y + Y + T u y + 4 u x xyt + xx ) + 4 9 y T + X y + 4Y y + Z 1 ( yt + X u x ] ρ u y + [ 4 T u 1 ut + 1 x T + xy T + xyx + xy + 1 9 y4 T + X y + Y y + Z 1, y + Z, xt + 1 ) ( ) ] ρ y T + yx + Y u x yt + X u y u ( + T u xx + ) [ ρ T + 4 u xx T u xy + 4 ( ) ] ρ yt + X yt + X u xx u xy [ + 5 T u x T u x + xt + y T + X y + Y xt + 1 ) ( ) ] ρ y T + yx + Y u xx yt + X u xy u x [ + 5 T u yy + 4 xt + 4 y T + 4X y + 4Y ( ) ] ρ u xt yt + X u xy u yy [ + u y T u y T + 4 xyt + xx + 4 9 y T + X y + 4Y y + Z 1, ( ) ( ) ( 1 yt + X u x yt + X u x xt + 1 ) y T + X y + Y u xy ( ) ] [ ρ yt + X u yy + 7 u y u T 5 u T 1 ut + 1 x T + xy T +xx y + xy + 1 9 y4 T + X y + Y y + Z 1, y + Z, xt + 1 ) y T + X y + Y u x xt + 1 ) y T + X y + Y u x ( ) ( ) ] ρ yt + X u y yt + X u y + T ρ = 0. (61) u The soluions J 1, J, and ρ of (59) (61) can be direcly found: ρ = f 0 ()K 1 ( 1,,,, 1 ), (6) J = [f 1 () + f ()y]k 1 ( 1,,,, 1 ) + f ()K ( 1,,,, 1 ), (6) J 1 = [f 4 () + f 5 ()x + f 6 ()y + f 7 ()y ]K 1 ( 1,,,, 1 ) +[f 8 () + f 9 ()y]k ( 1,,,, 1 ) + f 10 ()K ( 1,,,, 1 ), (64)
1144 SYMMETRY GROUP THEOREM TO THE LIN-TSIEN... VOL. 51 where K 1, K, and K are arbirary funcions of { 1,,,, 1 }, and f i, i = 0, 1,, 10 are funcions fixed by f 0 = T 1, (65) f 1 = XT, f = T T, f = T 4, (66) f 4 = Y T, f 5 = 1 T T, f 6 = X T, f 7 = 1 T T, f 8 = XT 7, f9 = T T 7, f10 = T 5, (67) wih he invarians being 1 = T y X1, (68) = T 1 x 1 T 4 T y T 4 Xy Y1, (69) = 1 T T x 10 81 T 8 y 4 T T y Y 5 T 8 yx T yy +T 5 xx T xy + 9 T 5 y 4 T T + 8 9 T 5 y X T + 4 9 T 5 y XT T yxx + 4 T 5 yxxt T xy T 1 9 T y 4 T T y X 0 7 T 8 y XT + 4 9 T 5 xy T +T 5 y XX 5 T 8 y X T + 4 T 5 y T Y + 4T 5 yxy +ut 1 + 1 Y, (70) 4 = u x T T 1 T x + 4 9 T 4 y T T 1 y T T 1 yx + 4 T 4 yxt + X T 4 Y T 1, (71) 5 = T u x y + T 1 xxt + 4 9 T 1 xyt 4 xyt + 4 9 T 1 y T T + 4 T 1 y X T + T 1 y XT 8 9 T y XT + T 1 yxx 4 T yx T + 4 T 1 yy T + XT 1 Y + Xu x xx 4 9 y T y X 4yY X T + u y T Y 16 81 T y T, (7)
VOL. 51 XI-ZHONG LIU AND JUN YU 1145 6 = 1 9 T 1 (9u T 9Z + 9u y X + 9Y u x T y 4 T x 18Y y 9Z 1 y 18Y x + T u 6X y + 6u y T y + T xu x 6xy T +T y u x + 9yu x X 18xX y), (7) 7 = u xx T T, (74) 8 = 1 9 T (9uxy T + 9u xx T X + 6u xx yt T + 4yT 1yT T 18T X + 6XT ), (75) 9 = 1 T (ux T + u xy X + u xy yt + u xx Y + u xx yx y T +u xx xt + u xx y T xt 6Y + u x T 6X y), (76) 10 = 1 7 T 4 (7uyy T + 54u xy T X + 6u xy yt T + 7u xx X T +6u xx yt XT + 1u xx y T T 108T Y 18X T + 6T Y T +18T T u x + 1xT T 6xT T 8y T + 1y T T T 6y T T 108yT X + 6yT X T 4yXT ), (77) 11 = u y T Z 1 T + u xy X XY + 4 9 u xyt + u xxt 4 yy T + u x T X + u yy T X + u x T X + 4 9 u xyy T xxt y XT yxx 4 9 y T T 4 y X T xt X y T X 4 9 y T T 4yT Y + u xy T Y +u xx XY + 1 T T u xy x + u xy yt X + 1 u xyy T T 4 xyt T + u xyt T + u xx yxx + u xxyt Y + 1 u xxy XT + u xxy T X + 4 u xyyxt 4 9 xyt T + 9 u xxy T T + 1 u xxxxt + 9 u xxxyt + T T u yy y + u xt T y + T T u y, (78)
1146 SYMMETRY GROUP THEOREM TO THE LIN-TSIEN... VOL. 51 where 1 = 1 9 T 1 (9Z, T 9Z 1 X + ut + 15u x yx T + 9u T + 9u yy X + 9u xx Y +15u T T 18Y Y + 18u y T X + 18u x T Y 6y T Y 6y T Y +ut T 18y T Y 6y T X 18xT Y T y 4 T T x T 14y X T 4y XT 18xXX 6xT Y 18y X X 0y Y T +9u xx y X 6y T X 6y T X y 4 T T y 4 T T x T T +u xx y 4 T + 18u xy XY + u xx x T + 4u yy y T 18yY X 6yXY xy T 18y XX Z T + u xx xy T T + 6u xx xyt X 9yT Z 1, +9u y T X + 9u x T Y 1xY T + 1u y XT + 9u x XX + 9u x T Y 18yY X 9yZ 1 T + u x xt + 8u y yt + 9u x yt X + 6u x y T T + 6u x xt T +1u y yt T 18xyT X + T u x y T + 18u x yt X + u x xt T +6u y yt T 6T xy T + 1u xy y T X + 6u xy y XT + 6u x yxt +4u xy xyt + 4u xy y T T + 6u xy xxt + 1u yy yxt + 6u xx xy T +6u xx y T X + 18u xx yy X 4xyX T 6xyX T 1xyXT +7u x y T T 1xy T T + 6u xx y Y T + 18u xy yxx + 1u xy yy T ), (79) X 1 = XT 5, (80) Y 1 = T 7 X + T 4 Y, (81) Y = Z 1, (8) Y = T 11 (15Y X T 6Y T + Z T 5X 4 XY T ). (8) To deermine he funcions of K 1, K, and K, we subsiue (6), (6), and (64) ino (58) which yields a complicaed equaion: J 1,x + J 1,u u x + J 1,ux u xx + J 1,uy u xy + J 1,u u x + J 1,uxx u xxx +J 1,uxy u xxy + J 1,ux u xx + J 1,uyy u xyy + J 1,uy u xy + J 1,u u x +J,y + J,u u y + J,ux u xy + J,uy u yy + J,u u y + J,uxx u xxy +J,uxy u xyy + J,ux u xy + J,uyy u yyy + J,uy u yy + J,u u y +ρ + ρ u u + ρ ux u x + ρ uy u y + ρ u u + ρ uxx u xx + ρ uxy u xy +ρ ux u x + ρ uyy u yy + ρ uy u y + ρ u u = 0. (84) To solve he complicaed equaion (84), we begin from he highes derivaives of u for K 1, K, and K being {u xxx, u xxy,, u } independen. Leing he coefficiens of
VOL. 51 XI-ZHONG LIU AND JUN YU 1147 {u xxx, u xxy,, u } be zero, we can ge a more simplified equaion. For example, he erm of u in (84) is where K i, n j case is T K 1,1 u, (85) = j K n. There is no nonrivial soluion for K j 1,1 0, hus he only possible i K 1,1 = 0, i.e., K 1 K 1 ( 1,,, 11 ). (86) Under (86), he coefficien of u y is T 8 (K,1 + K 1,11 ), (87) which leads o he only possible soluion K ( 1,,,, 1 ) = 1 K 1,11 + K 1 ( 1,,,, 11 ), (88) wih K 1 ( 1,,,, 11 ) being an undeermined funcion of he indicaed variables. Like he procedure o eliminae u and u y, vanishing he erms of u xxx, u xxy,, u yyy resuls in K 1 = (F 8 + F 7 + F 10 ) 11 + (F 1 7 + F 11 + F 9 ) 10 +(F 4 8 F ) 9 F 1 8 + (F 5 + F 6 ) 8 + F 8 7 + F 14, (89) K = (F 8 F 7 F 10 ) 1 + (F 1 7 F 9 F 11 ) 11 + 9F + (F 5 + F 1 8 ) 9 + F 7 7 + F 9 8 + F 1, (90) K = (F 10 + 8 F F 4 ) 1 + F 11 + (F 1 8 F 9 F 5 F 6 ) 11 + (F 1 9 F 9 ) 10 F 7 8 F 8 9 + F 1, (91) wih 14 equaions o be saisfied: F 1, 6 + F 7,5 F 9,4 = 0, F,4 + F 1,6 F,5 = 0, F 8, 6 + F 1,4 + F 7,1 + F 7, 5 = 0, F 4,4 + F 8,6 F, 5 F,1 = 0, F 9, + F 1,5 F 9, 4 + F 11, 6 = 0, F 11,6 + F, + F 10,5 + F, 4 = 0, F 10,1 F 4, 4 + F 14,6 F 10, 5 F 4, = 0, F 1,1 + F 1, 4 + F 1, 5 + F 14, 6 + F 1, = 0, F 5,5 F 11,4 + F 1, 4 + F 1, F, 6 + F 9,6 = 0, F 14,4 + F 5, 5 + F 5,1 + F 1,6 F 8, 4 F 8, + F 4, 6 = 0, F 8,5 F 7,6 + F 5,4 + F 1, 5 F, 6 + F 6,4 + F 1,1 = 0, F 4,5 + F, 5 F 10,4 + F,1 + F, 4 + F 6,6 + F 5,6 + F, = 0, (F 5, + F 6, ) 6 + F 9,1 F 7, 4 + F 1,5 + F 9, 5 + F 1,4 F 7, = 0, F 10, 6 (F 6, + F 5, ) 4 + F 1,6 + F 14,5 F 11,1 F 5, F 11, 5 F 6, = 0, (9)
1148 SYMMETRY GROUP THEOREM TO THE LIN-TSIEN... VOL. 51 where F i, i = 1,, 14 are funcions of { 1,,, 4, 5, 6 }. Forunaely he equaions are a linear sysem, which is sraighforward o solve. For simpliciy, we jus lis he final soluions: K 1 = (α,6 7 + α 5,6 α 1,5 6 8 ) 11 + (α 1,5 6 9 + (α 1,4 5 + α,5 + α 6,5 5 ) 7 +α 5,5 + α 7,5 5 + α 1, 5 + 4 α 1, 5 ) 10 + (α,6 8 + α,6 ) 9 +(α 1,4 5 α,5 α 6,5 5 ) 8 + (α 5,4 + α 8,5 α 1,1 5 4 α, 5 α 1, 5 + α,5 α, ) 8 + ((α 6, 5 + α, α 10,4 4 ) 5 + α 6, 4 +α 11,4 + 4 α 6, 4 + α,4 α 7,4 4 + α 6, + α,1 + α 6,1 5 + α 8,4 ) 7 +(α 7, 5 α 10, 4 + α 15, 4 α 10, 4 + α 10, + α 5, ) 5 + 4α 6, +(α 1, + α 11, α 7, 4 + α 8, + α, + α 6, ) 4 + α 17 + α 5,1 + α 7,1 5 +α, + α 8, + α 10,1 + α 15,1 + α 6, α 7, 4 + α 11, + α 1,, (9) K = (α 1,5 6 8 α,6 7 α 5,6 ) 1 + (α 1,5 6 9 + (α 1,4 5 α,5 α 6,5 5 ) 7 α 7,5 5 α 1, 5 4 α 1, 5 α 5,5 ) 11 + 9α,6 + ( 8 (α 1,4 5 + α,5 + α 6,5 5 ) 6 α 1, 6 + (α 6, 5 + α, ) 4 α 7,4 5 + α 9,6 α 1, α 5,4 + α 6, 5 +α 4,6 + α, ) 9 + α 4,5 8 + ((α 10,4 4 α, α 6, 5 α 1, 4 ) 6 + α 4,4 +α 9,4 + α 1,4 ) 7 + ((α 1, + α 10, 4 ) 4 α 10, α 5, α 7, 5 α 1, + α 10, 4 α 15, ) 6 + α 9, + α 4, + α 1, +(α 1, + α 14, + α 9, + α 4, ) 4 + α 14, α 16, + α 18, (94) K = (α 1,5 6 10 + α,6 8 α,6 ) 1 + 11α 1,5 6 + ( 8 (α 1,4 5 + α,5 + α 6,5 5 ) α,6 9 4 α 6, 5 + α 1, + α 7,4 5 + 6 α 1, 6 α 4,6 α 6, 5 α 9,6 α,5 + α 1,1 5 + 5 α 1, 5 α 8,5 ) 11 + ((α 1,4 5 α,5 α 6,5 5 ) 9 α 4,5 ) 10 + ((α, α 6, 5 + α 10,4 4 ) 5 α,1 α,4 α 6, 4 α 6, α 8,4 + α 7,4 4 α 6,1 5 4 α 6, 4 α 11,4 ) 9 + ((α 10,4 4 + α, + α 6, 5 +α 1, 4 ) 6 α 4,4 α 9,4 α 1,4 ) 8 + ( 5 α 1, 4 α 6, α 6, α 8, α, α 10,1 4 α 11, α 1, + α 1,1 + α 7, 4 ) 6 + (α 9, α 4, α 1, α 14, ) 5 α 4,1 α 1,1 α 14,1 + α 16,1 α 9,1, (95) wih α i (i = 1,,, 5) being arbirary funcions of { 1,,, 4, 5, 6 }, α i (i = 6, 7, 8) being arbirary funcions of { 1,,, 4, 5 }, α 9 being arbirary funcion of { 1,,, 4, 6 }, α i (i = 10, 11, 1) being arbirary funcions of { 1,,, 4 }, α 1, α 14, α 15 being arbirary funcions of { 1,, }, α 16, α 17 being arbirary funcions of { 1, }, and α 18 is a funcion of. Subsiuing (9) (95) ino (6) (64), we can obain he conservaion laws of he Lin- Tsien equaion associaed wih he Lie-Bäcklund generaor X 0. We have verified ha he conserved vecor (J 1, J, ρ) really saisfied Eq. (58).
VOL. 51 XI-ZHONG LIU AND JUN YU 1149 IV. CONCLUSION AND DISCUSSION In his paper, by applying he modified CK direc mehod, we se up Theorem 1, which shows he relaionship beween new exac soluions and old ones of he (+1)-dimensional Lin-Tsien equaion. Using he ransformaion relaions we hen ge he corresponding KMV symmery algebra and he Lie poin symmery which coincide wih he resul generaed from he sandard Lie approach. We generae he conservaion laws of he Lin-Tsien equaion relaed o he infinie dimensional KMV symmery group by use of he Lie-Bäcklund generaor up o he second order group invarians. The exisence of arbirary funcions of he group invarians proves he Lin-Tsien equaion has infiniely many conservaion laws which connec wih he general Lie poin symmery (7). Though he symmeries and conservaion laws are obained from he Lin-Tsien equaion, he conservaion laws we derived only depended on he symmery which may be possessed by many equaions. Acknowledgemens This work was suppored by he Naional Naural Science Foundaion of China under Gran Nos. 114718,117519, and he Naural Science Foundaion of Zhe Jiang Province of China (Gran No. Y7080455). References [1] M. J. Ablowiz and P. A. Clarkson, Solions, Nonlinear Evoluion Equaions and Inverse Scaering (Cambridge Universiy Press, Cambridge, 1991). [] M. J. Ablowiz and H. Segur, Solions and he Inverse Scaering Transform (SIAM, Philadelphia, 1981). [] R. X. Yao, X. Y. Jiao, and S. Y. Lou, Chin. Phys. B 18 181 (009). [4] J. H. Li and S. Y. Lou, Chin. Phys. B 17, 747 (008). [5] M. Jia, Chin. Phys. B 16, 154 (007). [6] Y. F. Wang, S. Y. Lou, and X. M. Qian, Chin. Phys. B 19, 0 (010). [7] W. F. Ames, Nonlinear Parial Differenial Equaions in Engineering (Academic Press, New York, 197). [8] G. W. Bluman and J. D. Cole Similariy Mehods for Differenial Equaions (Springer, New York, 1974). [9] L. V. Ovsiannikov Group Analysis of Differenial Equaions (NAUKA, 1978). [10] H. C. Ma, Chin. Phys. Le., 554 (005). [11] S. Y. Lou, J. Phys. A: Mah. Gen. 8, 19 (005). [1] X. Y. Tang, Y. Gao, F. Huang, and S. Y. Lou, Chin. Phys. B 18, 46 (009). [1] X. Y. Jiao and S. Y. Lou, Chin. Phys. B 18, 611 (009). [14] E. Noeher Nachr. König. Gesellsch. Wiss. Göingen, Mah-phys. Klasse, 5 (1918). [15] E. Bessel-Hagen, Mah. Ann. 84, 58 (191). [16] N. H. Ibragimov, Transformaion groups applied o mahemaical physics (Reidel, Boson,
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