Similarity Reduction And Exact Solutions Of Several Kinds Of Nonlinear Equations

Finding exact traveling wave solutions of nonlinear mathematical physics equations is one of the most important problems in soliton theory and mathematical physics. Thus many people have been made very hard efforts to solve nonlinear equations. But because of its complexity, people need to further study on the finding effective methods to solve large number of nonlinear partial differential equations in the future. However, it is quite important to get the specific methods for solving specific equations. Because some universal solving methods can be deduced from the methods constructed to solve the special equations. Also we can explain some new physical phenomena by using the special solutions for specific equations. Therefore, study on the special methods to solve nonlinear equations can give very good guideline for seeking new methods.In this paper, we obtained the similarity reduction equations, similarity solutions and exact solitary wave solutions of some nonlinear equations by using similarity reduction method(the direct reduction method established by Clarkson and Kruskal, the classical infinitesimal transformation, the non-classical infinitesimal transformation, etc) and the hyperbolic function method. This paper is divided into the following four chapters:The first chapter is the introduction part, in which a brief introduction of the development of the soliton theory, the history of similarity reduction method and the extended hyperbolic function method are presented. Finally introduces the main work of this paper.In the first section of the second chapter, we briefly introduce the CK direct reduction method. Then in the second section, the similarity reductions and similarity solutions of the SI-I equation and SI-II equation are given by using the CK direct method. In the third section, the similarity reduction equations and similarity solution of the fifth order XLY equation are given by use of the CK direct reduction method. We also obtained the rational solution and the soliton solution of the equation through discussion on the similarity reduction equations.The first section of third chapter introduces the classical infinitesimal transformation method. In the second section, by using the classical infinitesimal transformation method, the similarity reduction equations of the SI-I equation and SI-II equation are given. In the third section, the classical infinitesimal transformation method is used to obtain the similarity reduction equation for the fifth order XLY equation. In the fourth section, we briefly introduce the non-classical infinitesimal transformation method. In the fifth section, the non-classical infinitesimal transformation method is used to obtain the similarity reduction equations for the SI-I equation and SI-II equation. In the sixth section, the non-classical infinitesimal transformation method is used to present the similarity reduction equation of the fifth order XLY equation.The first section of chapter four introduces the hyperbolic function method(coupled Riccati equation method). In the second section, the hyperbolic function method is used to find the solitary wave solutions of some nonlinear equations. The third section introduces the simple idea of the extended hyperbolic function method. In the fourth section, by using the extended hyperbolic function method, the explicit exact solutions including the hyperbolic function solutions, triangle function solutions, travelling wave solutions and rational solutions are obtained.