Researches On Exact Solutions And Construction Of Soliton Equations

As one of two problems in soliton theory, it is the research point for the construction of in-tegrable soliton equations. In this dissertation, we proposed a method of constructing integrable soliton equations and give its applications by discussing some related open problems. The ap-proximate analytical solution is a important research topic, we give a method of construction of approximate analytical solutions and the applications are provided.In chapter one we present the historical development and current researches of soliton the-ory.In chapter two we introduce some basic knowledge about construction of soliton equations and exact solutions for the followed research.In chapter three an important model AC=BD is employed to reduce some nonlinear dif-ferential equations.In chapter four we obtain the main conclusions about construction of bilinear equations which satisfy the linear superposition principle and dispersion relation. As the applications of the method, we construct some new integrable equations.In chapter five we extend the conclusions in chapter four and present a method for con-structing supersymmetric bilinear equations which satisfy the linear superposition principle and dispersion relation. Then we give some examples.In chapter six we construct a multiparameter hommotopy method based on the concept in topology. Then the numerical examples show that theses themes provide good accuracy.In chapter seven we summarize the dissertation and give some topics for further research.