Symmetry Properties, Structure Of The Exact Solutions For Some Nonlinear Partial Differential Equations

With the rapid development of nonlinear science, Nonlinear Evolution Equation appeared. Nonlinear problems have been widely applied in almost all branches of physics and other natural sciences. The solution of a partial differential equation of the exact solution becomes the impor-tant research topic of scientists. Through the tireless efforts of mathematicians and physicists, They put forward a lot of convenient and effective method for solving the partial differential equation. For example, truncated Painleve expansion, Darboux transformation, Inverse scat-tering transformation, Backhand transformation and so on. In this article, we will based on symbolic computation, Lie symmetry group theory, put forward by the use of CK and further revised by professor sen-yue Lou CK direct method and the projective Riccati expansion method to study the symmetry group and the exact solutions of some nonlinear equationThe layout of this paper is as follows.· In chapter 1, we briefly introduce the development history of soliton and some methods to obtain the exact solutions of nonlinear evolution equations, then we also give the main topic.· In chapter 2, we introduce some basic concepts and properties.· In chapter 3, We briefly describe the CK direct method. By Professor sen-yue Lou improved CK direct method,we seek (3+1)-dimensional NEE finite symmetry transformation group and exact solutions. Then we use the projective Riccati expansion method to get some solutions of this equation. Some figures are given to illustrate some evolution features for these solutions.· In chapter 4, A consistent Riccati expansion (CRE) method is a effective method to con-struct the exact solutions of the nonlinear partial differential equations, by using the consis-tent Riccati expansion method, the consistent condition of Gardner equation is constructed. Solving the consistent equation, we find the soliton-cnoidal wave solutions.