The Study Of Exact Solutions Of Nonlinear 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 the solutions and pictures of the solutions of the nonlinear equations. Then, it is very important for the study and development of physics to solve the nonlinear partial differential equations and give the pictures of the 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 generalized auxiliary equation method and the Exp-function 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 study and 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 exact solutions by means of the generalized auxiliary equation method. Second, it is considered how to construct the exact solutions with the Exp-function 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 generalized auxiliary equation method and Exp-function method as well as the significance of studying the theory of soliton. In chapter 2, we give a description of the generalized auxiliary equation method and apply the method to the modified form of (1+1)-dimensional Degasperis - Procesi equation and the (2+1)-dimensional nonlinear dispersive Zakharov-Kuznetsov equation, so general solutions including arbitrary constants are obtained. In chapter 3, the Exp-function 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 (2+1)-dimensional nonlinear KP-BBM equation and the general types of combined KdV-Burgers equation with variable coefficients as examples. As a result, many new and more general exact solutions have been obtained. In chapter 4, some conclusions are given.