Multi-linear Variable Separation Approach For A Number Of Partial Differential Systems

With the development of modern mathematics, nonlinear science has infiltrated variety of territories which arise in life, for biology, economics, nature science, natural engineering technology, meteorological phenomenon, subsurface investigation which have widely applied. Recently, there are many methods for finding the numerical and exact solutions of nonlinear systems, for instance, Douboux transformation, inverse scattering method, CK direct method, and variable separation approach and so on. In this paper, we study the technique and method of variable separation apporach, and based on this method, we study a number of nonlinear equation with the help of the computer symbolic system Maple and Methematica.The thesis is arranged as follows:Chapter 1 Introduction. Overview the discovery, history and progress of soliton briefly and introduce several methods for solving nonlinear systems in brief. are introduced in this chapter.Chapter 2 We mainly introduce the multi-linear variable separation apporach. First, we overwise the development process and the current state. Second, we give out the steps that multi-linear variable separation appoarch solves nonlinear systems. Then we apply multi-linear variable separation approach to the (2+1)-dimensional Higher Order BK. At last,we introduce how to promote to the general multi-linear variable separation approach, getting the general multi-linear variable separation solutions.Chapter 3 Based on chapter2, we introduce another approach to solve nonlinear systems, then we set example to (2+1)-dimensional HBK, GNNV and Burgers system.Chapter4 Introduce the M-â…¨equation and give out its two sets of solutions of the gauge equivalent counterpart.Chapter5 Summary.