Application Of Three Wave Method And The Wronskian In Nonlinear Systems

Nonlinear physics develops fast with the development of nonlinear science.In nonlinear physics,simplified nonlinear evolution equations are often employed to describe the complex nonlinear physics system.The background,application Of nonlinear evolution equations in the field of physics and other and special properties of solitary wave solution is make the exact solutions of nonlinear evolution equations and soliton theory became a leading subject and hot interest in the study of nonlinear science.Beacause,The quantificational or the qualitative relations between physics quantities can be determined by solving the nonlinear equations.Besides of this,the first hand impression of the relations between physics quantities can be got by pictures of the solutions of the nonlinear equations.Then,it is very important for the development of physics to solve the nonlinear partial differential equations and give the pictures of the solutions.There are mainly two problems in the study of nonlinear partial differential equations.One is the complex calculation and reasoning in the process of looking for the exact solution and constructing soliton solution of equations,the second is innovation and promotion of method to solve the nonlinear partial differential equation.For the first problem,in recent years,the development of computers has simplify process of solving the equation and accelerated the research of nonlinear partial differential equation greatly.For the other problem,For the second problem,although a number of methods are proposed and developed to look for the exact solutions of nonlinear partial differential equations,unfortunately,not all these approaches are universally applicable for solving all kinds of nonlinear partial differential equations directly,many efficient methods have been established and developed to solve the nonlinear systems especially those integrable ones.For instance,The inverse scattering transformation,Darboux transformation,Backlund transformation,Tanh function method,similarity,Funetional variable separation approach,Bilinear method and Multilinear method,The homogeneous balance method,Classieal and nonclassieal Lie group approaehes,ClarksonKruskal's direct method,Deformation mapping method,Truncated Painleve expansion,Function expansion method,Homoclinic test technique,Double wave approach,Three wave approach,expand three wave methods and so on.New methods are emerging.Based on several nonlinear partial differential equation,this dissertation focused on the development of three wave method and the Wronskian technique in the application of solving nonlinear partial differential equation with the aid of computer algebraic system Maple.Therefore,some new exact solutions of equation have been obtained.The structure of the article is elucidated as follows.Chapter 1 is the part of introduction.It includes the discovery and recent developing character of soliton,studying of solutions of the nonlinear partial differential equations,the research and development of BKP equation and BLMP equation as well as the significance of studying the theory of soliton.In chapter 2,we give a description of the extended three-wave approach and apply it to(3+1)-dimensional BKP equations and the(3+1)-dimensional BLMP equations so three-wave solutions including several arbitrary parameters are obtained.In chapter 3,Hirota method based on the Wronskian technique is described for constructing more general exact solutions of nonlinear partial differential equations with the aid of symbolic computation,we apply the approach to(3+1)-dimensional BKP equations and the(3+1)-dimensional BLMP equations.As a result,many new and more general exact solutions have been obtained.In chapter 4,some conclusions are given.