Solutions for Two Kinds of Evolutional Equations 

Author  DangXiaoLan 
Tutor  ZhangYi 
School  Zhejiang Normal University 
Course  System theory 
Keywords  the generalized Schr(o ¨)dinger equation sixorder KdV equation Homotopy perturbation method Variational method the generalized Riccati mapping method θfunction B(a ¨)cklund transformation Hirota bilinear method 
CLC  O175.29 
Type  Master's thesis 
Year  2011 
Downloads  12 
Quotes  0 
To solve equations is very important in researching nonlinear evolutional equations, which is also a focus in research on the soliton theory. This article mainly concerns two integrable equations and three nonintegrable equations. The generalized nonlinear Schrodinger equation can be transformed into the standard Schrodinger equation, so we can get the solutions for the generalized nonlinear Schrodinger equations in the focusing and defocusing situations by using transformation. Moreover, we also obtain the periodic solutions in the focusing and defocusing situations by using the theta functions. Another integrable equation is sixorder KdV equation. Through the Hirota bilinear method, we get the N siliton solution and the Backlund transformation. Based the modified Backlund transformation, we obtain the limiting solutions of sixorder KdV equation. In addition, we study three nonintegrable equations. Using different methods, we get many solutions for the (3+1) dimensional equations.This article consists of four parts:The first chapter is an introduction, presenting an overview of the development process of soliton theory and some methods for researching the evolutional equations. The main work of this article is also introduced.In the second chapter, we first deduce the transformation for the nonlinear Schrodinger equation and transform it into the nonlinear standard Schrodinger equation. In focusing and defocusing cases, we obtain many forms of solutions. What’s more, the periodic solutions in terms of ellipticθfunctions are also driven. In addition, we present the bilinear form and the N soliton solution of the sixorder KdV equation. Furthermore, we work out the Backlund transformation and modified Backlund transforma tion. Based on the modified Backlund transformation, we also obtain the limiting solutions for this equation.The third chapter is mainly focused on the (3+1) dimensional equations. By using the homotopy perturbation method, variational method and the generalized Riccati mapping method, we get many solutions for the (3+1) dimensional equations.The last chapter gives concluding remarks and makes expectations for future work.