The Studies Of Some Typical Nonlinear Problems And The Soliton Perturbation Theory

Soliton theory is one of the important branch of nonlinear science, and Soliton perturbation problems is one of the important part of soliton theory.soliton exits in a great number of actual nonlinear systems ,So the study of soliton perturbation theory has been increasingly received extensive attention of many researchers. In this thesis, firstly ,Some typical nonlinear physical equations are derived from actual physical systems;and secondly a direct approach for soliton perturbations developed by professor Yan Jiaren etal is extended and improved,This approch relies on no tools of the inverse scattering transformation, and get rid of the proof of orthogonality for eigenfunctions, it can be employed to deal not only with integrable but also with unintegrable systems, and is very simple and easy to understand.This technique provided powerful tool for theoretical and experimetal worker.This thesis consist of four charpter. Chapter one includes mainly following three parts. Firstly,we briefly introduce the nonlinear phenomina relent to physics. Secondly, we concisely recall the formation and development of soliton perturbation theory. And we emphasis on two kinds of representative soliton perturbation theories, at the same time we point out some merits and defects of these theories respectively. Thirdly, we briefly exponde significance of the research for soliton perturbation theory. In chapter two, we derived some typical nonlinear physical equations from actual physical problems, and found their soliton solutions.In chapter three,we dealed with some nonlinear perturbation equation by use of Yan's direct approach for soliton perturbation. In chapter four, we developed a direct approach for soliton perturbation based on the Green function, This approch get rid of the proof of orthognality for eigenfunctions in a direct approch for soliton perturbation developed by Yan Jiaren etal.