Exact Solutions Of The Soliton Equations

There are many important subjects in soliton theory.Among them,to find exact solutions to soliton equations is an essential but important sub-ject.Various methods have been develped to search solutions for solition equations.For example,Hirota bilinear method and Wronskian technique are two important tools to deal with soliton problems.The former can be used to effectively construct the N-soliton solution in the form of Nth-order polynomial in N exponentials for a large class of NLEEs.While the latter provides a simple and straightforward way of verifying the validity of the N-soliton solution by virtue of properties of the Wronskian determi-nant.One part of this thesis is deriving the exact solutions for some soliton equations by virtue of the anterior two method.Until now,much attention has been paid to the problem of soliton equations with self-consistent sources(SESCSs).SESCSs constitute an important class of integrable equations and have extensive physical appli-cations.Some methods have been developed to study effectively SESCSs and fruitful results have been obtained.Recently,Hu Xingbiao and Wang Hongyan propose a algebraic method-source generation procedure,to dis-cuss SESCSs.One of the advantages of this new approach is that SESCSs and their soliton solutions can be generated simultaneously from the pro-cedure.The other part of this thesis is to construct SESCSs via source generation procedure.The specific work consists of three parts:In chapter 2,we study the complex KdV equation.Two kinds of solu-tions are obtained by means of the Wronskian technique and the Backlund transformation.Besides that,we will show that they are equivalent. In chapter 3,we discuss the exact solutions and some integrable prop-erties to the variable-coefficient modified Korteweg-de Vries equation.First, we derive the N-soliton solutions by means of Hirota bilinear method and Wronskian technique.Second,Backlund transformation and Lax pairs for this equation are given,for explaining the integrability of it.At last,we show the interaction of some solitary waves via some pictures.In chapter 4,we discuss the nonisospectral mKP equation with self-consistent sources(mKPESCS).First,Two types of Grammian solutions to the nonisospectral mKP equat.ion are derived with the help of the Pfaf-fian derivative formulae and Pfaffian identity.Second,we construct the mKPESCS by applying the source generation procedure,and the solutions of the equation we obtained can then be given.