The Study Of Lie Symmetry, The Exact Solutions And Conservation Laws For Some Nonlinear Partial Differential Equations

It is one of the important ways to solve the exact solutions of partial differential equations by reducing differential equations through Lie symmetry method. And in solution theory, conservation law of partial differential equations plays an important role in discussing the linearization, integrality, numerical calculations and other aspects of solutions, conservation laws are closely related to the existence of solutions.In this paper, by using the software package Mathematic of notations algorithm to analyze Lie symmetry method 、symmetry classification and optimal system of some nonlinear partial differential equations. Some new exact solutions of some nonlinear partial differential equations are achieved. And construct the conservation laws by using Lie- B¨acklund algorithm and achieve the new conservation laws.Chapter 1 is an introduction to review the theoretical background and development of Lie symmetry method, one dimensional optimal system and conservation laws, and the method of constructing the conservation laws of partial differential equation and notations involved in the paper is given.Chapter 2 concentrates on applying the Lie symmetry method in the modified c Kd V equations. Firstly, the symmetry of equations is achieved by using Lie symmetry method,which is allowed in the modified c Kd V equations, one dimensional optimal system and the exact solutions of equations are given. In addition, the new conservation laws of the modified c Kd V equations are obtained from the Lie- B¨acklund method.Chapter 3 concentrates on applying the Lie symmetry method in the Hirota-Satsuma equations. Firstly, the symmetry of equations is achieved by using Lie symmetry method,which is allowed in the Hirota-Satsuma equations, one dimensional optimal system and the exact solutions of equations are given. In addition, the new conservation laws of the HirotaSatsuma equations are obtained from the Lie- B¨acklund method.Chapter 4 gives detailed study on applying the Lie symmetry method in the KaupKupershmidt equation. Firstly, the symmetry classification of equation are achieved by using Lie symmetry method, one dimensional optimal system and the exact solutions of equations are given. In addition, the new conservation laws of the Kaup-Kupershmidt equation are obtained from the Lie- B¨acklund method.In chapter 5, the summary and discussion of this dissertation are given, as well as the outlook of future work is discussed.