The Study Of Exact Solutions Of Some Partial Differential Equations

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 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.In this dissertation,Multilinear variable separation approach are studied,and employed to solve the(2+1)-dimensional Burgers equations and the(2+1)-dimensional KdV equations separately.At last,the new periodic waves and the localized excitations of the(2+1)-dimensional Burgers equations and the(2+1)-dimensional KdV equations are obtained by selecting the arbitrary functions properly in their solutions.Many research topics,such as searching for exact explicit solutions,multi-soliton solution,et al.,often involve a large amount of tedious algebra auxiliary reasoning or calculations which can become unmanageable in practice.In recent years,the development of symbolic computation accelerates the research of nonlinear partial differential equation greatly.Many new methods for constructing exact solutions of nonlinear partial differential equations are proposed.This dissertation mainly studies some aspects of nonlinear partial differential equations with the aid symbolic computation,which include searching exact solutions of some nonlinear partial differential equations by means of the auxiliary equation method proposed in recent years.In order to illustrate the validity and the advantages of the method we choose some nonlinear partial differential equations as examples.As a result,many new and more general exact solutions have been obtained.As mentioned above,the solutions of nonlinear equations have vital significance to the physics development,many mathematicians and physicians have done massive work in this aspect,but actually discover no method possible to solve all equations. Therefore in view of each class of equations,people are always exploring new methods to solve them.This article includes two main parts.First,it is studied on how to seek the complex solutions by means of the MLVSA.Second,it is considered how to construct the exact solutions with the auxiliary equation method.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 multilinear variable separation approach and auxiliary equation method as well as the significance of studying the theory of soliton.In chapter 2,we give a description,of the generalized multilinear variable separation approach and apply the MLVSA to(2+1)-dimensional Burgers equations and the(2+1)-dimensional KdV equations so general solutions including two arbitrary functions are obtained.In chapter 3,by choosing these arbitrary functions appropriately,a class of double periodic wave solutions for the equations are given and the interaction properties of the periodic waves by virtue of the numerical method are studied,reveal that some of them are nonelastic and some are completely elastic.Furthermore,these results are visualized by using their figures.In chapter 4,the generalized auxiliary equation method is described for constructing more general exact solutions of nonlinear partial differential equations with the aid of symbolic computation.In order to illustrate the validity and the advantages of the method,we choose the modified generalized Camassa-Holm equation as an example.In chapter 5, we expand the auxiliary equation method to high order equations,ie.,the fifth-order KdV equation and the seventh-order KdV equation.As a result,many new and more general exact solutions have been obtained.