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The Weierstrass elliptic function expansion method and its applications in nonlinear wave equations. (English) Zbl 1142.35603
Summary: Based on the close relationship between the Weierstrass elliptic function Weierstrass ${\frak p} (\xi ; g_{2}, g_{3})$ and nonlinear ordinary differential equation, a Weierstrass elliptic function expansion method is developed in terms of the Weierstrass elliptic function instead of many Jacobi elliptic functions. The mechanism is constructive and can be carried out in computer with the aid of computer algebra ({\tt Maple}). Many important nonlinear wave equations arising from nonlinear science are chosen to illustrate this technique such as the new integrable Davey-Stewartson-type equation, the $(2 + 1)$-dimensional modified KdV equation, the generalized Hirota equation in $2 + 1$ dimensions, the Generalized KdV equation, the $(2 + 1)$-dimensional modified Novikov-Veselov equations, $(2 + 1)$-dimensional generalized system of modified KdV equation, the coupled Klein-Gordon equation, and the $(2 + 1)$-dimensional generalization of coupled nonlinear Schrödinger equation. As a consequence, some new doubly periodic solutions are obtained in terms of the Weierstrass elliptic function. Moreover solitary wave solutions and singular solitary wave solutions are also given as simple limits of doubly periodic solutions. These solutions may be useful to explain some physical phenomena. The algorithm is also applied to other many nonlinear wave equations. Moreover we also present the general form of the method.

##### MSC:
 35Q55 NLS-like (nonlinear Schrödinger) equations 33E05 Elliptic functions and integrals 35C05 Solutions of PDE in closed form 35Q53 KdV-like (Korteweg-de Vries) equations 37K10 Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies
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