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# Multi-linear Variable Separation Approach and Localized Excitations in Nonlinear Systems

Author: DuYanYan
Tutor: LiuGuanTing
School: Inner Mongolia Normal
Course: Applied Mathematics
Keywords: Multi- linear variable separation method Nonlinear systems Localized excitations
CLC: N93
Type: Master's thesis
Year: 2008
Downloads: 94
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### Abstract

 Linear theory is improving, nonlinear science has become the focus of research to flourish in various research fields. Can not be avoided in the course of the study will encounter a variety of nonlinear equations, and for these non-linear equations, nonlinear science research will undoubtedly become the key, also nonlinear study difficulty. Unlike linear equations, due to the failure of the linear superposition principle, there is no way to give the general solution of the nonlinear system essentially. Although a particular solution can be used to get one or several methods, but a method usually can not get the types of the particular solution. Therefore, no uniform method for solving nonlinear systems. Through the efforts of many scientists, it has been established and developed an effective method for solving nonlinear systems, multi-linear variable separation method belongs to one of them, it is to achieve a true sense of the separation of variables. So far, the multi-linear variable separation method has been successfully solving a large class of 21-dimensional nonlinear system and 11-dimensional and 31-dimensional nonlinear system. Multi-linear variable separation method has been successfully applied to a differential differential systems. This thesis is focused on the multi-linear variable separation method of variables separation techniques, resulting in a new multi-linear separation of variables solution by selecting the appropriate function of any new localized excitations. In addition, the multi-linear variable separation method can also be further promote the general multi-linear variable separation approach, resulting in some general nonlinear system of multi-linear separation of variables solution, this solution contains a more low-dimensional variable separation function. Select the proper function of any new localized excitations of the nonlinear system mode. Now the main content of this article is summarized as follows: Chapter 1 is an introduction, introduces the discovery and study of the history of the solitary wave, summed up the situation of the current study, a brief mathematical study of nonlinear systems, and the papers research work arrangements. The second chapter introduces the first multi-linear variable separation method of the course of development and current developments, followed by summary discusses the general steps of the multi-linear variable separation method for solving nonlinear systems. Contain any function (generally contain p and q) variable separation solutions obtained by the application of multi-linear variable separation method is the focus of our research. Then we will be a multi-linear variable separation method applied to the 21-dimensional Boiti-Leon-Pemponelli (BLP) equation and the 21-dimensional long-wave dispersion equation to discuss its new variable separation solutions and localized excitation mode. Select appropriate new solution contains arbitrary function, can be quite abundant form of soliton solutions, in general, if p and q are solvable model of multi-linear variable separation method for a variety of single-valued function can be produce many single-valued localized excitations, such as arched soliton peak sub reverse soliton kink. Chapter multi-linear variable separation method further promote the general multi-linear variable separation method, assumed that the further promotion of the upcoming multi-linear variable separation method of separation of variables, resulting in a nonlinear system of general linear separation of variables solution, this solution contains more low-dimensional variable separation function. By choosing the proper function of any get rich localized excitations. Chapter IV, gives the main results of this paper, the idea of ??some of the future research work.

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