The Theory Of Soliton Perturbations Of Nonlinear Schr(?)dinger Equation And It's Application

Soliton theory is one of the important branches of nonlinear science. It has crescent application in many fields. The soliton perturbation problem is an important part of the soliton theory. It exists in a large number of real nonlinear systems and can be roughly divided to two kinds. One is based on the inverse scattering transformation(IST) which has important learning value. But this technique is inconvenient to those who are not familiar with 1ST. Another is the direct method where the squared Jost solutions are employed as the basis for perturbation expansion after soliton eqation been lineared. They are just applicable to integral systems. Professor Jiaren Yan, who is my thesis supervisor, had developed a direct approach of the perturbation thoery based on separating variable technique, which is applicable to both integrable and unintegral systems. It is more simple and convenient in method and calculation. I tackle the perturbation problem of the nonlinear Schrodinger equation because of its importance. At the same time, it enriched the soliton perturbation thoery of Professor Yan and offered important examples. Next, I studied the stability of bright solitons in Bose-Einstein condensate based on the direct approach.This thesis consists of five chapters. The first chapter has two parts. First, we briefly introduce the development history of the soliton and discuss some general approaches to deal with the soliton perturbation problems, and point out some drawbacks of these approaches. Secondly, we present the general procedure of our approach and its major characteristics. In the second chapter, we establish the perturbation theory for the nonlinear Schrodinger equation, and study its specific perturbation. We will explain the two different approaches, which are based on the direct approach through the work that we have done. In the third chapter we briefly introduce the formation and development of Bose-Einstein condensation's theory and it's experiments. Then we derive the nonlinear Gross-Pitaevaskii equation that satisfies the condensate macroscopic wave function. At the end of this chapter, we discuss the theoretical studies...