Solutions for Two Kinds of Evolutional Equations
|School||Zhejiang Normal University|
|Keywords||the generalized Schr(o ¨)dinger equation six-order KdV equation Homotopy perturbation method Variational method the gen-eralized Riccati mapping method θfunction B(a ¨)cklund transformation Hirota bilinear method|
To solve equations is very important in researching nonlinear evolu-tional equations, which is also a focus in research on the soliton the-ory. This article mainly concerns two integrable equations and three non-integrable 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 fo-cusing and defocusing situations by using transformation. Moreover, we also obtain the periodic solutions in the focusing and defocusing situa-tions by using the theta functions. Another integrable equation is six-order 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 six-order KdV equa-tion. In addition, we study three non-integrable 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 de-velopment 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 non-linear 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 six-order 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 equa-tions. 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.