Symmetry and Exact Solutions of (2+1)-Dimensional Generalized Sasa Satsuma Equation via a Modified Direct Method

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1 Commun. Theor. Phys. Beijing, China pp c Chinese Physical Society and IOP Publishing Ltd Vol. 51, No., June 15, 2009 Symmetry and Exact Solutions of 2+1-Dimensional Generalized Sasa Satsuma Equation via a Modified Direct Method LÜ Chang-Cheng1 and CHEN Yong 1,2, 1 Nonlinear Science Center and Department of Mathematics, Ningbo University, Ningbo 15211, China 2 Shanghai Key Laboratory of Trustworthy Computing, East China Normal University, Shanghai 20002, China Received May 19, 2008 Abstract In this paper, first, we employ classic Lie symmetry groups approach to obtain the Lie symmetry groups of the well-known 2+1-dimensional Generalized Sasa Satsuma GSS equation. Second, based on a modified direct method proposed by Lou [J. Phys. A: Math. Gen L129], more general symmetry groups are obtained and the relationship between the new solution and known solution is set up. At the same time, the Lie symmetry groups obtained are only special cases of the more general symmetry groups. At last, some exact solutions of GSS equations are constructed by the relationship obtained in the paper between the new solution and known solution. PACS numbers: 02.0.Jr Key words: classic Lie symmetry groups approach, modified CK s direct method, generalized Sasa Satsuma equation 1 Introduction Lie symmetry is one of the most powerful methods in every branch of natural science especially in integrable systems. [1,2,] To find the Lie point symmetry group of a nonlinear system, there is a standard method to find symmetry algebras and groups for almost all the known integrable systems. In previous paper, [4] Clarkson and Kruskal CK introduced a simple direct method to derive symmetry reductions of a nonlinear system without using any group theory. For many types of nonlinear systems, those symmetry gained by CK s direct method can also be generated via the classical and nonclassic Lie group approach. [5] In previous paper, [] the conditional similarity reducation solution can also be obtained by the modified CK s direct method. These facts hint that there is a simple method to find generalized symmetry groups for many types of nonlinear equations. In this paper, we consider the Generalized Sasa Satsuma GSS equation, [7] which has this following form: 1 q t + q xxx + q x U + qu x = 0, 1 2 r t + r xxx + r x U + ru x = 0, 2 U y qr x = 0, and is a generalization of the Sasa Satsuma equation introduced recently: [8] q t + q xxx + q x U + qu x = 0, 4 U y q 2 x = 0, 5 where q is the complex physical field and U is the real potential for Eqs. 1. When r is taken as the complex conjugate of q, Eqs. 1 become the Eqs. 4 and 5. This paper is arranged as follows: In Sec. 2, first, with the method of classic Lie group, the Lie point symmetry of GSS equation can be obtained, then by using the modified direct method, we get a relationship between the new solution and known solution of the GSS equation. Taking some special case, the Lie symmetry of the GSS equation also can be gained, by solving the corresponding characteristic equation, then the reduced equations can be obtained. In Sec., some explict solutions of GSS equation are presented via choosing different parameters. Finally, some conclusions and discussions are given in Sec Symmetry and Reduction of GSS Equation In this section, we look for the symmetry of Eqs. 1 by using the classical Lie method. Let us consider a one-parameter Lie group of infinitesimal transformation x x + ɛξx, u, t, q, r, U, y y + ɛηx, u, t, q, r, U, t t + ɛτx, u, t, q, r, U, q q + ɛφ 1 x, u, t, q, r, U, r r + ɛφ 2 x, u, t, q, r, U, U U + ɛφ x, u, t, q, r, U, with a small parameter ɛ, the corresponding vector field associated with the above group of transformations to GSS equation can be written as V = ξ x + η y + τ t + φ 1 q Supported by the National Natural Science Foundation of China under Grant No , Shanghai Leading Academic Discipline Project under Grant No. B412, National Natural Science Foundation of China under Grant No , Program for Changjiang Scholars and Innovative Research Team in University under Grant No. IRT074 and K.C. Wong Magna Fund in Ningbo University chenyong@nbu.edu.cn

2 974 LÜ Chang-Cheng and CHEN Yong Vol φ 2 r + φ U, where ξ, η, τ, φ 1, φ 2, and φ are the funcations with respect to {x, y, t, q, r, U}, the prolongation of the vector field is pr V = V + φ x 1 + φ t 1 + φ xxx 1 + φ x 2 q x q t q xxx r x + φ t 2 + φ xxx 2 + φ x + φ y, q t r xxx U x U y where φ x 1 = D xφ 1 ξq x ηq y τq t + ξq xx + ηq xy + τq xt, φ t 1 = D t φ 1 ξq x ηq y τq t + ξq xt + ηq yt + τq tt, φ xxx 1 = D xxx φ 1 ξq x ηq y τq t + ξq xxxx + ηq xxxy + τq xxxt, φ x 2 = D x φ 2 ξr x ηr y τr t + ξr xx + ηr xy + τr xt, φ t 2 = D t φ 2 ξr x ηr y τr t + ξr xt + ηr yt + τr tt, φ xxx 2 = D xxx φ 2 ξr x ηr y τr t + ξ xxxx + ηr xxxy + τr xxxt, φ x = D x φ ξu x ηu y τu t + ξu xx + ηu xy + τu xt, φ y = D yφ ξu x ηu y τu t + ξu xy + ηu yy + τu yt. By using these extensions and applying the prolongation to the GSS equation, the infinitesimal criteria for the invariance of Eqs. 1 under the group is given by pr V i j =0 = 0 i, j = 1, 2,. After substituting Eqs. 1 into Eq. for i, j = 1, 2,, comparing the coefficients of linearly independent expressions of {q, r, U} and let their derivatives be zero q t, r t, U y and their derivatives have been cancelled by i i = 1, 2,, we obtain various constraint equations. And solving these equations, one can obtain F ξ = 1 tx + F 4 t, η = F 2 y, τ = F 1 t, φ 1 = 1 q F 2 y + F 1 t + F y, F φ 2 = 1 t + F y r, φ = F 1 t 18 x + F 4 t 2 F 1 t U. If we want to obtain the relation between the new solution and known solution of GSS equation, one has to solve the following ODEs d x dɛ = ξ x, ȳ, t, q, r, Ū, dȳ dɛ = η x, ȳ, t, q, r, Ū, d t dɛ = τ x, ȳ, t, q, r, Ū, 7 d q dɛ = φ 1 x, ȳ, t, q, r, Ū, d r dɛ = φ 2 x, ȳ, t, q, r, Ū, dū dɛ = φ x, ȳ, t, q, r, Ū, 8 with initial solution conditions x ɛ=0 = x, ȳ ɛ=0 = y, t ɛ=0 = t, q ɛ=0 = q, r ɛ=0 = r, Ū ɛ=0 = U. 9 Solving the Eqs. 7 9 is difficult, [9] because the equations inclued some arbitrary functions, such as F 1 t, F 4 t, F 2 y, F y. If we want to get the transformation between the new solution and known solution of GSS equation, one may use the modified direct method. For the GSS equation, in order to search the transformation between the new solution and known solution, it is enough to seek the symmetry reductions in a simple form [10] q = α + βqξ, η, τ, 10 r = γ + θrξ, η, τ, 11 U = ζ + φwξ, η, τ, 12 where α = αx, y, t, β = βx, y, t, γ = γx, y, t, θ = θx, y, t, ζ = ζx, y, t, φ = φx, y, t and ξ, η, τ are functions of {x, y, t} to be determined later, under the transformations {x, y, t, qx, y, t, rx, y, t, Ux, y, t} {ξ, η, τ, Qξ, η, τ, Rξ, η, τ, Wξ, η, τ}, and satisfy the same PDEs Q τ + Q ξξξ + Q ξ W + QW ξ = 0, 1 R τ + R ξξξ + R ξ W + RW ξ = 0, 14 W η QR ξ = Substituting Eqs along with Eqs into Eqs. 1 and eliminating Q τ, R τ, W η and their higher order derivatives by means of the Eqs Let the coefficients of U, V, W and their derivatives be zero, we arrive at some PDEs which are omitted in this paper. By solving the above PDEs, we have the following results: ξ = f 1 t 2 x f 2 tf 1 tdt, η = f yf 4 ydy, τ = f 1 t dt, α = 0, γ = 0, β = f yf 1 t, θ = f 4 yf 1 t, f ζ = 1 tx f 1 t + f 2t, φ = f 1 t 4.

3 No. Symmetry and Exact Solutions of 2+1-Dimensional Generalized Sasa Satsuma Equation 975 In summary, by using Eqs , the following theorem holds: Theorem 1 If Qξ, η, τ, Rξ, η, τ, Wξ, η, τ are the solutions of the GSS equation, then the GSS equation also has solutions as follow: qx, y, t = f yf 1 tqξ, η, τ, rx, y, t = f 4 yf 1 trξ, η, τ, Ux, y, t = f 1 t f 1 t x + f 2t + f 1 t 4 Wξ, η, τ. To obtain the Lie point symmetry group obtained from Theorem 1, we restrict: f 1 t = 1 + ɛft, f 2 t = ɛf 2 t, f y = 1 + ɛf y, f 4 y = 1 + ɛf 4 y with an infinitesimal parameter ɛ, then equations in Theorem 1 can be written as: q = Q + ɛδq, r = R + ɛδr, U = W + ɛδw. Furthermore if setting F ft = 1 t, f 2t = Ḣ4t, f y + f 4 y = Ġ4y, we also obtain the symmetry 1 tx δq = + H 4 t Q x + G 4 yq y 1 t + F 1 tq t + δr = δu = 1 tx + H 4 t + F 1 tr t + 1 t 1 tx + H 4 t + F 1 tw t F 1 tx + F y Q, 1 R x + G 4 yr y + F 4 y R, 17 W x + G 4 yw y Ḣ4t F 1 t W. 18 If one set some constraint conditions, the Eqs are equal to the vector field obtained by classic Lie point group. The solutions of Eqs are obtained by solving the characteristic equation, [5] then one obtain the similarity form of the solution as q = F 1 x, y, t, Φ 1 X, Y, r = F 2 x, y, t, Φ 2 X, Y, U = F x, y, t, Φ X, Y, 19 where X and Y are similarity variables, the others Φ 1 X, Y, Φ 2 X, Y, Φ X, Y are dependent variables. Substituting Eqs. 19 into Eqs. 1, then we can obtain the reduction equations of GSS equation. By setting F 1 t = t, H 4 t = t, G 4 y = y, F y = 0, F 4 y = 1 as an example, the invariant surface conditions become x + t Q x + yq x + tq x + 1 Q = 0, 20 x + t R x + yr x + tr x + 7 Q = 0, 21 x + t W x + yw x + tw x W = The corresponding characteristic equation is dx x/ + t = dy y = dt t = dq 1/ = dr 7/ dw = 1/ 2/W. 2 Solving the characteristic Eq. 2, we obtain q = t 7/ Φ 1 X, Y, r = t 1/ Φ 2 X, Y, U = 1 5 t + t 2/ Φ X, Y, 24 5x t2 X =, Y = y 5t 1/ t. 25 Substituting Eqs. 24 and 25 into Eqs. 1, the reduction of GSS equation read 7Φ 1 + 2XΦ 1X + Y Φ 1Y Φ 1XXX Φ 1X Φ 18Φ 1 Φ X = 0, 2 Φ 2 + 2XΦ 2X + Y Φ 2Y Φ 2XXX Φ 2X Φ 18Φ 2 Φ X = 0, 27 Φ Y Φ 1X Φ 2 Φ 2X Φ 1 = As we see, the reduced GSS equation are still complex and to solve the Eqs is difficult. However, the above results and discussion set up the relations between new and the known solution, which can construct new solutions. In addition, the more general symmetry group are obtained, which include the classical point symmetry group. Solution of 2+1-Dimensional GSS Equation In the paper, [7] the author solve the GSS equation by the Painlevé approach and the final solutions can be given in terms of arbitrary functions as follow φ2x φ 1y q = 2Fφ 1 + φ 2, 29 r = F φ 2x φ 1 + φ 2, 0 φ 2 2x U = 2φ 1 + φ φ 2xx 2φ 1 + φ 2 φ 2t + φ 2xxx φ 2x + φ2 2xx 8φ 2, 1 2x where φ 2 is an arbitrary function of {x, t} and φ 1 and F are arbitrary functions of y. Also the Eqs have

4 97 LÜ Chang-Cheng and CHEN Yong Vol. 51 the same form solutions: φ2ξ φ 1η Q = 2Fφ 1 + φ 2, 2 R = F φ 2ξ φ 1 + φ 2, W = φ 2 2ξ 2φ 1 + φ φ 2ξξ 2φ 1 + φ 2 φ 2τ + φ 2ξξξ φ 2ξ + φ2 2ξξ 8φ 2, 4 2ξ where φ 2 is an arbitrary function of {ξ, τ} and φ 1 and F are arbitrary functions of η. Here, applying the symmetry group Theorem 1 on Eqs. 2 4, one can obtain some solutions when the seed solutions of GSS equation are given, φ2ξ φ 1η q = f yf 1 t 2Fφ 1 + φ 2, 5 r = f 4 yf 1 t F φ 2ξ φ 1 + φ 2, U = f 1 t f 1 t x + f 2t f 1 t 4 φ 2 2ξ 2φ 1 + φ 2 2 φ 2ξξ 2φ 1 + φ 2 + φ 2τ φ 2ξξξ φ 2ξ where ξ = f 1 t 2 x f 2 tf 1 tdt, η = f yf 4 ydy, and τ = f 1 t dt are given in Sec. 2. In order to get some explicit results, we choose the arbitrary functions φ 1 and φ 2 as simple as where a 0, a 1, a, k 1, ω 1 are arbitrary constants. φ 1 = a e η, φ 2 = a 0 + a 1 e k 1ξ ω 1 τ, φ2 2ξξ 8φ 2, 7 2ξ Fig. 1 Plots of the GSS equation given by Eqs. 5a, b, and 7c with parameter choice Eqs. 8 and 9 all at time t = Figure 1 displays the solutions structure of Eqs. and 7, with the parameter selections a 0 = 20, a 1 = 20, a = 1, k 1 =, ω 1 = 2, k = 1 2, 8

5 No. Symmetry and Exact Solutions of 2+1-Dimensional Generalized Sasa Satsuma Equation 977 f 1 t = cost 1/, f 2 t = 0, f y = 1, f 4 y = y, Fη = 1. 9 Fig. 2 Plots of the GSS equation given by Eqs. 5a, b, and 7c with the parameter choice Eqs. 40 and 41 all at time t = Figure 2 shows the solution structure of Eqs. 5 7, for the parameter choice a 0 = 20, a 1 = 2, a = 1, k 1 = ω 1 = 2, k = 1 2, 40 f 1 t = cost 1/, f 2 t = 0, f y = tanhy, f 4 y = sechy, Fη = For generating the types of periodic wave interaction solution of the GSS equation, the arbitrary functions also can be set as: N φ 1 = [a 1i sn α 1i k 1i η 1i, m 1i + a 2i cn α 2i k 1i η 1i, m 2i + a i dn α i k 1i η 1i, m i ], i=1 φ 2 = b 0 + N [b 1i sn β 1i ζ 1i, m 1i + b 2i cn β 2i ζ 2i, m 2i + b i dn β i ζ i, m i ], i=1 where ζ ij = l ij ξ + ω ij τ and l ij, ω ij, a ij, b ij, α ij, β ij, b 0 are arbitrary constants. 4 Summary and Conclusion In summary, based on the classic Lie groups approach and a modified direct method, we investigate the wellknown 2+1-dimensional Generalized Sasa Satsuma GSS equation. As a result, the more general symmetry groups obtained by a modified direct method include the classical Lie point group by classical method; a transformation of GSS equations solution between the new solution and known solution is set up. By means of the transformation and the known solution, some exact solutions of GSS equations are constructed. It is necessary to point out that the reduced equation by the direct method maybe or not simpler than the original equation. If the reduced equation is relative simply, we can directly use the classic Lie groups approach to solve it and obtain the solution of target equation by

6 978 LÜ Chang-Cheng and CHEN Yong Vol. 51 the transformation between the new solution and the solution of reduced equation. The method reported here can be applied in principle to all nonlinear systems including both integrable and non-integrable ones and more details will be reported in future. References [1] MA Hong-Cai and LOU Sen-Yue, Commun. Theor. Phys [2] P.J. Olver, Application of Lie Group to Differential Equations, Springer, Berlin 198. [] G.W. Bluman and J.D. Cole, Similarity Methods for Differential Equations, Applied Mathematical Sciences, Vol. 1, Springer, Berlin [4] P.A. Clarkson and M.D. Kruskal, J. Math. Phys [5] Sen-Yue Lou, J. Math. Phys [] Sen-Yue Lou, Xiao-Yan Tang, and Ji Lin, J. Math. Phys [7] R. Radha and S.Y. Lou, Physica Scripta [8] C. Gilson, J. Hietarinta, J. Nimmo, and Y. Ohta, Phys. Rev. E [9] D. David, N. Kamran, D. Levi, and P. Winternitz, J. Math. Phys [10] H.C. Ma, Commun. Theor. Phys