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Zbl 1039.35029
Fan, Engui
Multiple travelling wave solutions of nonlinear evolution equations using a unified algebraic method. (English)
[J] J. Phys. A, Math. Gen. 35, No. 32, 6853-6872 (2002). ISSN 0305-4470

Summary: A 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 tanh methods, the Jacobi elliptic function method 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 coupled Hirota-Satsuma, coupled Schrödinger-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.

MSC 2000:: *35G20 General theory of nonlinear higher-order PDE
33F10 Symbolic computation of special functions
35Q53 KdV-like equations
37K10 Completely integrable systems etc.

Cited in: Zbl 1062.37094

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