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Multiple travelling wave solutions of nonlinear evolution equations using a unified algebraic method 

注意:本論文已在 JOURNAL OF PHYSICS A: MATHEMATICAL AND GENERAL 35 (2002) 6853–6872 》發表
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Engui Fan
Institute of Mathematics, Key Laboratory for Nonlinear Mathematical Models and Methods,
Fudan University, Shanghai 200433, People’s Republic of China

Received 2 April 2002, in final form 7 June 2002
Published 2 August 2002

AbstractA new direct and unified algebraic method for constructing multiple travelling wave solutions of general nonlinear evolution equations is presented and implemented in a computer algebraic system. Compared with most of the existing tanhmethods, the Jacobi elliptic functionmethod or other sophisticated methods, the proposed method not only gives new and more general solutions, but also provides a guideline to classify the various types of the travelling wave solutions according to the values of some parameters. The solutions obtained in this paper include (a) kink-shaped and bell-shaped soliton solutions, (b) rational solutions, (c) triangular periodic solutions and (d) Jacobi and Weierstrass doubly periodic wave solutions. Among them, the Jacobi elliptic periodic wave solutions exactly degenerate to the soliton solutions at a certain limit condition. The efficiency of the method can be demonstrated on a large variety of nonlinear evolution equations such as those considered in this paper, KdV–MKdV, Ito’s fifth MKdV, Hirota, Nizhnik–Novikov–Veselov, Broer–Kaup, generalized coupledHirota–Satsuma, coupled Schr¨odinger–KdV, (2 + 1)-dimensional dispersive long wave, (2 + 1)-dimensional Davey– Stewartson equations. In addition, as an illustrative sample, the properties of the soliton solutions and Jacobi doubly periodic solutions for the Hirota equation are shown by some figures. The links among our proposed method, the tanh method, extended tanh method and the Jacobi elliptic function method are clarified generally.

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