B(?)cklund Transformation And Conservation Laws For Modified Forms Of Camassa-Holm And Degasperis-Procesi Equations And Its Exact Solutions

Backlund transformation and conservation laws play an important role in the study of nonlinear partial differential equations. In this paper, we carry on a detailed introduction to Backlund transformation and conservation laws for modified Camassa-Holm and Degasperis-Procesi equations, and explain the important roles of Backlund transformation and conservation laws for exact solutions of modified Camassa-Holm and Degasperis-Procesi equations.This paper is organized as follows:In chapter one, we briefly retrospect the historical background and developing processes of soliton, completely generalize the main methods of solving nonlinear partial differential equation brought forward from home and abroad, and also introduce to the point the aim of research and the primary contents of this paper.In chapter two, we introduce in detail the expression of modified Camassa-Holm and Degasperis-Procesi equation, the background for its revision, the physical significance for the equation,and important results of modified Camassa-Holm and Degasperis-Procesi equations which have obtained by famous mathematician until now.In chapter three, we first introduce Backlund transformation's background, theo-retical meaning, practical function and the relationship of Backlund transformation method and other methods for finding solutions of nonlinear evolution equations. Then we presente the explanation for finding the modified Camassa-Holm equation's Backlund transformation, and we obtaine more solutions by using Backlund transfor-mation and a simple solution of modified Camassa-Holm equation.In chapter four, we present the explanation for finding the modified Degasperis-Procesi equation'Backlund transformation, and we obtain more solutions by using Backlund transformation and a simple solution of modified Degasperis-Procesi equa-tion. And we compare these solutions to currently existing solution.In chapter five, we introduce in detail the method for finding conservation laws of modified Camassa-Holm and Degasperis-Procesi equations, and we obtaine conserva-tion laws of modified Camassa-Holm and Degasperis-Procesi equations by using the method which is found by Anco and Bluman.In chapter six, we introduce in detail the method of Hirota for finding the solu-tions of nonlinear partial differential equations. And we also use Hirota's method and Maple 10 to find the solutions of the modified form of Camassa-Holm and Degasperis-Procesi equations.Finally, we make a summary of this paper and look ahead of research orientation in future.