Exact Soliton-like Solutions Of Several Kinds Of Nonlinear Evolution Equations

The problem for solving nonlinear evolution equations is an oldand important topic. Although, mathematicians and physicists have donevery successful research in this area, but a general method for solvingnonlinear evolution equations has not been established yet because of thecomplexity of nonlinear differential equations. Fortunately, there aremany effective methods for constructing exact solutions for nonlinearequations in soliton theory, such as the homogeneous balancemethod[4-8]、 inverse scattering method[21]、 bilinear transformationmethod[33]、auxiliary equation method[47-55]、Darboux transformationmethod[60-66]etc. This dissertation mainly attempt to solve severalnonlinear evolution equations by use of the auxiliary equation methodand Darboux transformation method. The dissertation consists of threechapters:The first chapter owes to introduction of the soliton theory, itsdevelopment and summary of the method for solving nonlinear evolutionequations. At the end of this chapter we shall introduce the main work ofthis dissertation.In the first section of the second chapter, the general process of theauxiliary equation method is described. In the second section, thesolutions of the auxiliary equation given in [47] is reclassifiedaccording to the independent relations. And then by use of these solutions and the symbolic computation system, we shall present some new exactsoliton like solutions of the variable coefficients combined KdV equation,variable coefficients Burgers equation and variable coefficients (3+1)-dimensional ZK equation.The third chapter, the concept of the Darboux transformationmethod is introduced at first. And then, a new assumption on Darbouxmatrix for Dirac system and HBK equations is used to obtain theirDarboux transformation as well as new soliton solutions.