LIE SYMMETRY GROUP OF (2+1)-DIMENSIONAL JAULENT-MIODEK EQUATION

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1 M, H.-C., et l.: Lie Symmetry Group of (+1)-Dimensionl Julent-Miodek THERMAL SCIENCE, Yer 01, ol. 18, No. 5, pp LIE SYMMETRY GROUP OF (+1)-DIMENSIONAL JAULENT-MIODEK EQUATION by Hong-Ci MA *, Ai-Ping DENG, nd Yo-Dong YU Deprtment of Applied Mthemtics, Donghu University, Shnghi, Chin Originl scientific pper DOI: 10.98/TSCI10557M In this pper, we consider system of (+1)-dimensionl non-liner model by using uxiliry eqution method nd Clrkson-Kruskl direct method which is very importnt in fluid nd physics. We construct some new exct solutions of (+1)-dimensionl non-liner models with the id of symbolic computtion which cn illustrte some ctions in fluid in the future. Key words: symmetry, exct trveling wve solutions, Julent-Miodek eqution, Clrkson-Kruskl direct method Introduction Non-liner evolution eqution plys n importnt role in pplied mthemtics nd physics. In recent yers, vrious effective methods hve been developed to find the exct solutions of non-liner prtil differentil equtions. These methods include tnh function method [1], generlized hyperbolic function method [], homogeneous blnce method [], Jcobi elliptic function expnsion method [], exponentil function method [5], uxiliry eqution method [6], Clrkson Kruskl (CK) direct method [7, 8] nd so on. The purpose of this work is to generlize the work mde in [9, 10]. We pply this method to the (+1)-dimensionl Julent-Miodek eqution which ssocites with energy-dependent Schrudinger potentil nd hs mny interesting chrcters. It is n importnt model in fluid nd physics. New exct trveling wve solutions for (+1)-dimensionl Julent-Miodek eqution In [11], Geng et l. developed some non-liner models generted by the Julent- Miodek hierrchy [1]. Wu [1] gve the N-soliton solution of the first model by using the Hirot biliner method. Liu et l. [1] discussed the bifurction nd exct trvelling wve solutions of the third one. Wzwz [15] obtined the Multiple kink solutions nd multiple singulr kink solutions of the third model. We hve studied the second model in [16] nd obtined the multiple kink solutions. In this pper, we shll discuss the following (+1)-dimensionl Julent-Miodek eqution: wt = ( wxx w ) x x wyy + wx x wy (1) By substituting w = u x into (1), we cn omit the integrl term in eq. (1): * Corresponding uthor; e-mil: hongcim@hotmil.com

2 M, H.-C., et l.: Lie Symmetry Group of (+1)-Dimensionl Julent-Miodek 158 THERMAL SCIENCE, Yer 01, ol. 18, No. 5, pp uxt + uxxxx uxuxx + uyy + uxxuy () 16 To seek the trvelling wve solution of eq. (), we introduce: u = U( ξ ), ξ = x+ dy+ et () where d nd e re rbitrry constnts. Substituting () into eq. () nd integrting once with respect to ξ nd setting the integrtion constnt equl to zero, one hs: ξξξ ξ ξ ξ ξ U + 16dU 8U + e U + 6U e= 0 By introducing nother trnsformtion: we cn rrive: ( ξ ) = () U ξ ξξ + 16d 8 + e + 6 e (5) Blncing the liner terms of the highest order with the non-liner terms yields the leding order m = 1. If we propose tht hs the form: ( ξ) = + z( ξ) (6) 0 1 where 0, nd 1 re constnts to be determined, nd z(ξ) express the solutions of [9, 10]: d( z ξ ) z ( ξ ) bz ( ξ) cz ( ξ) dξ = + + (7) Substituting eqs. (7) nd (6) into eq. (5) nd setting the coefficients of z i (ξ) (i, 1,, ) to zero, we obtin following set of non-liner lgebric equtions: d + e + 6e c= d + e + 1e b + 6e Solving this set of lgebric equtions with the id of Mple, we obtin: = d e, 0, b= e1, c= 1 (8) 1 1 8d e e = + + e e 18 d, 0 e e 18 d, ± + = ± b = 1 e e 18 d, c 1 ± + = (9) Substituting eq. (8) with z(ξ) in [10] into eq. (6) gives the exct solution of eq. (5):

3 M, H.-C., et l.: Lie Symmetry Group of (+1)-Dimensionl Julent-Miodek THERMAL SCIENCE, Yer 01, ol. 18, No. 5, pp d e 1 d + e sech ξ 1( ξ ) = d e e d + e tnh ξ d e 1 d + e csch ξ ( ξ ) = d e e d + e coth ξ where ξ = x + dy + et, nd 1, d, nd e re rbitrry constnts nd d + (/)e < 0,, d + e 1 d + e sec ξ ( ξ ) = d + e e d + e tn ξ d + e 1 d + e csc ξ ( ξ ) = d + e e d + e cot ξ where ξ = x + dy + et, nd 1, d, nd e re rbitrry constnts nd d + (/)e > 0. By using eq. (), we obtin: 1 1 ln 1tnh d e U = e + e1 + 1 d e tnh d e ξ 1 ln tnh d e ξ U 1 = ln e tnh d + e + ξ 1

4 M, H.-C., et l.: Lie Symmetry Group of (+1)-Dimensionl Julent-Miodek 1550 THERMAL SCIENCE, Yer 01, ol. 18, No. 5, pp d e tnh d e 1 ln tnh d e U ln e1 1 d e tn d e = ξ 1 U ln tg d e ln e1tn d e = d + e nd solution of eq. (1) with w = u x. Substituting eq. (9) with z(ξ) in [11] into eq. (6) yields the exct trveling wve solution of eq. (5) s the sme wy s the former cse. Symmetry group trnsformtion of the (+1)-dimensionl Julent-Miodek eqution nd its new exct solutions The ppliction of Lie theory hs been plying n importnt role since it ws demonstrted by Lie [17]. In 1989, Clrkson nd Kruskl [18] developed the CK direct method tht cn be used to find symmetry reduction. Recently, Lou nd M [19] introduced modified CK direct method to obtin symmetry trnsformtion group of given PDE. To get symmetry trnsformtion of eq. (), we suppose: u = α + βu( ξ, η, τ) (10) where α = α(x, y, t), β = β(x, y, t), ξ = ξ(x, y, t), η = η(x, y, t), τ = τ(x, y, t) re rbitrry functions of x, y, nd t to be determined by restricting U(ξ, η, τ) to stisfy the sme eqution s u under the trnsformtion {u, x, y, t} {U, ξ, η, τ}. Restrict U to stisfy the sme eqution s u, sy: 1 Uξτ + Uξξξξ UξUξξ + Uηη + UξξUη (11) 16 Substituting eq. (10) with eq. (11) into eq. () nd setting the coefficients of U nd its derivtives equl to zero, we rrive t some equtions to be determined. After some tedious clcultion, we hve: 8 τ 8 p ξ = τ x y y+ q, η = τ y+ p, β = 1, τ = τ( t) (1) 1/ tt t / t / 1/ t 9 τ t τt t τtt tτtt t τtt / 5/ / t τt t t t px xy p y 8 p y y α = τ 9 7 τ 9τ 81 τ + τttt y qt y 16 ptt y mt () (1) 1/ / 81 τ τ 9 τ t t t where p = p(t), q = q(t), τ = τ(t) nd m(t) re the rbitrry function of t. By using eq. (1), we hve:

5 M, H.-C., et l.: Lie Symmetry Group of (+1)-Dimensionl Julent-Miodek THERMAL SCIENCE, Yer 01, ol. 18, No. 5, pp u = α + βu( ξ, η, τ) where α, β, ξ, η, re τ re determined by eq. (1) nd (1). We lso get the symmetry: σ ( U) = y ftt nt y+ h+ ft xux + n+ yft Uy + fut + s ftt xy y f xn yh + y n 81 9 The generl elements of Lie lgebr cn be written s: ttt t t tt = ( f) + ( h) + ( n) + ( s) ( f ) = y ftt + ftx + yft + f + fttxy + y fttt 9 x y t 9 81 U ( ) = x h h yh t U 8 16 ( n) = nty + n + xnt + y ntt x y 9 U () s s = U To our knowledge, the symmetry groups of eq. () hve not been studied in literture. Summry We hve presented the uxiliry eqution method nd Lie symmetry trnsformtion to construct more generl exct solutions of NLPDE with the id of Mple. We hve successfully obtined mny new exct trveling wve solutions which my be useful for describing certin non-liner physicl phenomen in fluid. It is shown tht the lgorithm cn be lso pplied to other NLPDE in mthemticl physics. Hence, the further study is needed. Acknowledgments The work is supported by the Ntionl Nturl Science Foundtion of Chin (project No ), the Fund of Science nd Technology Commission of Shnghi Municiplity (project No. ZX ) nd the Fundmentl Reserch Funds for the Centrl Universities. References [1] Prkes, E. J., Duffy, B. R., An Automted Tnh-Function Method for Finding Solitry Wve Solutions to Non-Liner Evolution Equtions, Computer Physics Communictions, 98 (1996),, pp [] Go, Y. T., Tin, B., Generlized Hyperbolic-Function Method with Computerized Symbolic Computtion to Construct the Solitonic Solutions to Non-liner Equtions of Mthemticl Physics, Computer Physics Communictions, 1 (001),, pp [] Wng, M. L., et l., Appliction of Homogeneous Blnce Method to Exct Solutions of Non-liner Equtions in Mthemticl Physics, Physics Letters A, 16 (1996), 1, pp

6 M, H.-C., et l.: Lie Symmetry Group of (+1)-Dimensionl Julent-Miodek 155 THERMAL SCIENCE, Yer 01, ol. 18, No. 5, pp [] Liu, S., et l., Jcobi Elliptic Function Expnsion Method nd Periodic Wve Solutions of Non-liner Wve Equtions, Physics Letters A, 89 (001), 1, pp [5] He, J.-H., Wu, X.-H., Exp-Function Method for Non-liner Wve Equtions, Chos, Solitons & Frctls, 0 (006),, pp [6] Sirendoreji, New Exct Trvelling Wve Solutions for the Kwhr nd Modified Kwhr Equtions, Chos, Solitons & Frctls, 19 (00), 1, pp [7] M, H. C., Generting Lie Point Symmetry Groups of (+1)-Dimensionl Broer-Kup Eqution vi Simple Direct Method, Communictions in Theoreticl Physics, (005), 6, pp [8] M, H. C., et l., Lie Symmetry Groups of (+1)-Dimensionl BKP Eqution nd Its New Solutions, Communictions in Theoreticl Physics, 50 (008),, pp [9] Sirendoreji, Sun, J., Auxiliry Eqution Method for Solving Non-liner Prtil Differentil Equtions, Physics Letters A, 09 (00), 5, pp [10] M, H. C., et l., The Auxiliry Eqution Method for Solving the Zkhrov-Kuznetsov (ZK) Eqution, Computers & Mthemtics with Applictions, 58 (009), 11, pp [11] Geng, X. G., et l., Qusi-Periodic Solutions for Some (+1)-Dimensionl Integrble Models Generted by the Julent-Miodek Hierrchy, Journl of Physics A: Mthemticl nd Generl, (001), 5, pp [1] Julent, M., Miodek, I., Non-liner Evolution Equtions Associted with Energy-Dependent Schrodinger Potentils, Letters in Mthemticl Physics, 1 (1976),, pp. -50 [1] Wu, J., N-Soliton Solution, Generlized Double Wronskin Determinnt Solution nd Rtionl Solution for (+1)-Dimensionl Non-liner Evolution Eqution, Physics Letters A, 7 (008), 1, pp [1] Liu, H., Yn, F., The Bifurction nd Exct Trvelling Wve Solutions for (+1)-Dimensionl Nonliner Models Generted by the Julent-Miodek Hierrchy, Interntionl Journl of Non-liner Science, 11 (011), 1, pp [15] Wzwz, A. M., Multiple Kink Solutions nd Multiple Singulr Kink Solutions for (+1)-Dimensionl Non-liner Models Generted by the Julent Miodek Hierrchy, Physics Letters A, 7 (009), 1, pp [16] Lie, S., Lectures on Differentil Equtions with Known Infinitesiml Trnsformtions, Teubner, Leipzig, Germny, 1891 [17] Clrkson, P. A., Kruskl, M. D., New Similrity Reductions of the Boussinesq Eqution, Journl of Mthemticl Physics, 0 (1989), 10, pp [18] Lou, S., M, H. C., Non-Lie Symmetry Groups of (+1)-Dimensionl Non-liner Systems Obtined From Simple Direct Method, Journl of Physics A: Mthemticl nd Generl, 8 (005), 7, pp. L19-L17 [19] M, H. C., et l., Symmetry Trnsformtion nd New Exct Multiple Kink nd Singulr Kink Solutions for (+1)-Dimensionl Non-liner Models Generted by the Julent-Miodek Hierrchy, Communictions in Theoreticl Physics, 59 (01),, pp Pper submitted: Mrch 10, 01 Pper revised: April 0, 01 Pper ccepted: July 1, 01