The Theory Of Soliton Perturbations Of Nonlinear Schr(?)dinger Equation And It's Application In Bose-Einstein Condensate And Optical Fiber

Soliton theory is one of the important problems of physics. It has extensive application in many fields. The theory of soliton perturbation have significant meaning in practical application. It can be divided roughly into two types. One is the theory of perturbation based on the inverse scattering transformation(IST), which is essential method in searching the solution of soliton, so it have important academic meaning, however, the method is not easily understood and grasped because of its complicated line of thinking. Another is the direct theory of perturbation, the method of the squared Jost solutions constructed as the basis of perturbation is the most universal in several symmetric ways. The method can't completely get rid of the dependence to the IST because that the squared Jost appear as the particular solution in the process of the IST. It is the " quasi-direct method " in strict terms. The above two methods are just applicable to integral systems, not the unintegral systems. My hierophant of professor Yan Jia-Ren developed a new direct method of the soliton perturbation based on separating variable technique from 1996 to 1998. It is not only easily comprehended and caught, but also more simple and convenient in calculation. It is the most important that the method cast off the effect of the IST. The method is applicable to the integral and unintegral systems. Recently, he deal with successfully the perturbation theory of the dark soliton. In my paper, the method is dealt with the nonlinear Schrodinger equation with several perturbation terms. I obtained rich meaning results through exploring and analyzing the dark soliton of optical fiber and Bose Einstein-Condensates.This paper consists of six chapters. The first chapter has two parts. First, we briefly introduce sorts of soliton equation and the types of soliton, several methods of perturbation theory of soliton. I analyze the strongpoint and drawbacks of these approaches. In the second chapter, I introduced in detail the perturbation theory of the dark soliton for the nonlinear Schrodinger equation, giving the ideas and process of treatment. In the third chapter we briefly introduce the Bose-Einstein condensation's theory and it's experiments,and the nonlinear Gross-Pitaevaskii equation. In the fourth chapter, I study thestability of the dark soliton in the Bose-Einstein Condensates and analyze the main properties of dark soliton. In the fifth chapter, I introduced the stability of optical soliton and studied the properties of the dark soliton in optical fiber. I summarize my work and discuss its prospect in Chapter Six.