Applications of the Jacobi elliptic function method to special-type nonlinear equations. (English) Zbl 1005.35063

Fan, Engui; Zhang, Jian

Phys. Lett., A 305, No. 6, 383-392 (2002).

Summary: The Jacobi elliptic function method with symbolic computation is extended to special-type nonlinear equations for constructing their doubly periodic wave solutions. Such equations cannot be directly dealt with by the method and require some kinds of pre-possessing techniques. It is shown that soliton solutions and triangular solutions can be established as the limits of the Jacobi doubly periodic wave solutions. The different Jacobi function expansions may lead to new Jacobi doubly periodic wave solutions, triangular periodic solutions and soliton solutions. In addition, as an illustrative sample, the properties for the Jacobi doubly periodic wave solutions of the coupled Schrödinger-KdV equation are shown with some figures.

MSC:

35L70	Second-order nonlinear hyperbolic equations
35C10	Series solutions to PDEs
35Q53	KdV equations (Korteweg-de Vries equations)
35Q55	NLS equations (nonlinear Schrödinger equations)

Keywords:

symbolic computation; doubly periodic wave solution; coupled Schrödinger-KdV equation

Software:

MACSYMA

References:

[1]	Malfliet, W, Am. J. phys., 60, 650, (1992)
[2]	Malfliet, W; Herman, W, Phys. scr., 54, 563, (1996)
[3]	Parkes, E.J; Duffy, B.R, Comput. phys. commun., 98, 288, (1996)
[4]	Duffy, B.R; Parkes, E.J, Phys. lett. A, 214, 271, (1996) · Zbl 0972.35528
[5]	Parkes, E.J; Duffy, B.R, Phys. lett. A, 229, 217, (1997) · Zbl 1043.35521
[6]	Hereman, W, Comput. phys. commun., 65, 143, (1991)
[7]	Hereman, W; Nuseir, A, Math. comput. simulation, 43, 13, (1997)
[8]	Gao, Y.T; Tian, B, Comput. math. appl., 33, 115, (1997)
[9]	Fan, E.G, Phys. lett. A, 277, 212, (2000)
[10]	Fan, E.G, Z. naturforsch. A, 56, 312, (2001)
[11]	Fan, E.G; Zhang, J; Hon, Y.C, Phys. lett. A, 291, 376, (2001)
[12]	Li, Z.B; Liu, Y.P; Wang, M.L, Acta math. sinica, 22, 138, (2002)
[13]	Yao, R.X; Li, Z.B, Phys. lett. A, 297, 196, (2002)
[14]	Liu, S.K; Fu, Z.T; Liu, S.D; Zhao, Q, Phys. lett. A, 289, 69, (2001)
[15]	Fu, Z.T; Liu, S.K; Liu, S.D; Zhao, Q, Phys. lett. A, 290, 72, (2001)
[16]	Fan, E.G; Hon, Y.C, Phys. lett. A, 292, 335, (2002)
[17]	Parkes, E.J; Duffy, B.R; Abbott, P.C, Phys. lett. A, 295, 280, (2002)
[18]	Ford, L.R, Automorphic functions, (1951), Chelsea New York · Zbl 1364.30001
[19]	Akhiezer, N.I, Elements of theory of elliptic functions, (1990), American Mathematical Society Providence, RI · Zbl 0694.33001
[20]	Ablowitz, M.J; Segur, H, Solitons and the inverse scattering transformation, (1991), SIAM Philadelphia, PA · Zbl 0299.35076
[21]	Leung, K.M; Mills, D.L; Riseborough, P.S; Trullinger, S.E, Phys. rev. B, 27, 4017, (1983)
[22]	Salerno, M; Quintero, N.R, Phys. rev. E, 65, 025602, (2002)
[23]	Lou, S.Y; Ni, G.J, Phys. lett. A, 140, 33, (1989)
[24]	Gani, V.A; Kudryavtsev, A.E, Phys. rev. E, 60, 3305, (1999)
[25]	Hirota, R, J. math. phys., 14, 805, (1973)
[26]	Maccari, A, J. math. phys., 39, 6547, (1998)
[27]	Yoshinaga, T; Wakamiya, M; Kakutani, T, Phys. fluids A, 3, 83, (1991)
[28]	Rao, N.N, Pramana J. phys., 29, 109, (1997)
[29]	Singh, S.V; Rao, N.N; Shukla, P.K, J. plasma phys., 60, 551, (1998)
[30]	Roy Chowdhury, A; Dasgupta, B; Rao, N.N, Chaos solitons fractals, 9, 1747, (1997)
[31]	Hase, H; Satsuma, J, J. phys. soc. jpn., 57, 679, (1988)
[32]	Davey, A; Stewartson, K, Proc. R. soc. London ser. A, 101, 101, (1974)
[33]	Tajiri, M; Takeuchi, K; Arai, T, Phys. rev. E, 64, 056622, (2001)
[34]	Gu, C.H; Hu, H.S; Zhou, Z.X, Darboux transformations in soliton theory and its geometric applications, (1999), Shanghai Sci. Tech. Publishing
[35]	Lou, S.Y; Lu, J, J. phys. A, 29, 4209, (1996)
[36]	Paul, S.K; Roy Chowdhury, A, J. nonlinear math. phys., 5, 349, (1998)