Symmetry And Exact Solutions Of The Nonlinear Partial Differential Equation

In this dissertation, under the instruction of the ideas of the mathematics mechanization and the AC=BD theory proposed by Prof. Zhang Hongqing and by means of symbolic com-putation software Maple, we study the applying of symmetry method in solving nonlinear partial differential equations(PDEs).The symmetry method related to group theory is an effective tool of investigating nonlinear PDEs.It plays an important role in mathematical and physical field. With the symmetry method, similarity reduction, order reducing, group classification, construction of conservation law and exact solutions of PDEs can be obtained. Extending the classical symmetry(Lie symmetry) method, some more general methods are obtained, such as generalized symmetry method, conditional symmetry method, generalized conditional symmetry method and so on.The generalized conditional symmetry method can be used to solve some nonlinear PDEs with initial/boundary-value problems, which is hard for the classical symmetry method. The main work is done as follows:In chapter 1,the ideas of mathematics mechanization, history and development of soli-tons theory and the symmetry method are introduced. The related investigation and develop-ment at home and abroad are also presented.In chapter 2,the AC=BD theory, the basic contents and ideas of C-D pair are intro-duced.In chapter 3,finite symmetry transformation groups and exact solutions of integrable CDGKS equation are studied. Starting from the Lax pair, the approved direct method is applied to obtain CDGKS equation's symmetry transformation group and exact solution. The related Lie point symmetry is presented in further.In chapter 4, the solving of the quasi-linear evolution equation, which admits gen-eralized conditional symmetry, with initial-value problem is investigated. The completed group classification, reduction of Cauchy problems and construction of exact solutions for the quasi-linear evolution equation are presented. And some relations between the general-ized conditional symmetry method and AC=BD theory are found.