Sobolev Periodic Solutions of Nonlinear Wave Equations in Higher Spatial Dimensions

2009 ◽  
Vol 195 (2) ◽  
pp. 609-642 ◽  
Author(s):  
Massimiliano Berti ◽  
Philippe Bolle
Keyword(s):  
Periodic Solutions ◽  
Nonlinear Wave ◽  
Wave Equations ◽  
Nonlinear Wave Equations ◽  
Spatial Dimensions

A REDUCTION mKdV METHOD WITH SYMBOLIC COMPUTATION TO CONSTRUCT NEW DOUBLY-PERIODIC SOLUTIONS FOR NONLINEAR WAVE EQUATIONS

2003 ◽  
Vol 14 (05) ◽  
pp. 661-672 ◽  
Author(s):  
ZHENYA YAN
Keyword(s):  
Periodic Solutions ◽  
Nonlinear Wave ◽  
Symbolic Computation ◽  
Wave Equations ◽  
Kdv Equation ◽  
Soliton Solutions ◽  
Transformation Method ◽  
Mkdv Equation ◽  
Nonlinear Wave Equations ◽  
Cubic Nonlinear Schrödinger Equation

Firstly twenty-four types of doubly-periodic solutions of the reduction mKdV equation are given. Secondly based on the reduction mKdV equation and its solutions, a systemic transformation method (called the reduction mKdV method) is developed to construct new doubly-periodic solutions of nonlinear equations. Thirdly with the aid of symbolic computation, we choose the KdV equation, the coupled variant Boussinesq equation and the cubic nonlinear Schrödinger equation to illustrate our method. As a result many types of solutions are obtained. These show that this method is simple and powerful to obtain more exact solutions including doubly-periodic solutions, soliton solutions and singly-periodic solutions to a wide class of nonlinear wave equations. Finally we further extended the method to a general form.

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