The Extended Conservation Laws And Soliton Solutions Of Nonlinear Evolution Equation

In this dissertation, under the guidance of mathematics mechanization and by means of computer algebraic system software, some problems of constructing conservation laws and solitary solutions of nonlinear evolution equations are discussed and some extended conservation laws of nonlinear evolution equations and exact solutions of (2+1)-dimensional variable coefficient nonlinear Schrodinger equation are constructed. They are realized on symbolic computation system Maple. The main work is as follows:Chapter 1 introduces the related development of mathematical physics mechanization, from the origins to the development at this stage. Classical integrable systems and the conservation law of nonlinear evolution equations are also presented.Chapter 2 introduces the theory of AC=BD and its application in seeking the solution of differential equation.In Chapter 3 we introduce the basic theory of extended conservation law and how to constructure the extended symmetries and extended conservation laws of given partial differential equations with Noether theorem. Firstly, the associated equations for evolution equations are matched. Thus the extended equations and Lagrangian are obtained; Secondly, a computational method to produce extended symmetries and extended conservation laws of given partial differential equation(s) are suggested; Finally, as applications of the method given in this paper, extended conservation laws of three evolution equations included Drinfel'd-Sokolov-Wilson equation, Benjamin-Bona-Mahony equation and KdV-Burgers-Kuramoto equation are constructured. The validity of the method is explained at the same time.In Chapter 4, with AC=BD Theory, the (2+1)-dimensional variable coefficient nonlinear Schrodinger equation can turn to the Klein-Gordon equation. Then the new soliton solutions of the variable coefficient nonlinear Schrodinger equation can be obtained by the solutions of the Klein-Gordon equation.