The Study Of Solutions To Some Nonlinear Evolution Equations

With the development of science and technology,the study of Nonlinear Evolution Equations(NLEEs) in different physics fields has attracted more and more attention.Solitons as an important branch of the Nonlinear Science,have been well studied and widely applied,which is of important physics significance. Finding the solutions of the NLEEs,especially,exact solutions,is an old and important research both in theory and application.Up to now,there are many kinds of methods to obtain the solutions of the NLEEs such as homogeneous balance method,bilinear B(a|¨)cklund transformation method,Hirota method, Tanh-function method.This paper takes the NLEEs as a foundation,studies several kinds of significant methods to solve the NLEEs,improves the solution method and find new exact solutions.The following are the basic contents of this paper:In chapter one,we first introduce the history and development of the soliton,and then by means of several examples explain two methods—similarity transformation and homogeneous balance method. In chapter two,we study the Hirota method,which is developped in 1970. We introduce the bilinear operator,the special properties of the bilinear operator and the transformations.As an example,we find an exact solution for the(2+1)-KdV equation.Chapter three focuses on a new bilinear B(a|¨)cklund transformation.Based on the known bilinear B(a|¨)cklund transformation of(2+1)-KdV equation,we find a new bilinear B(a|¨)cklund transformation and then obtain new soliton solutions.Chapter four as the focus of this paper in which we study another determinant solution for nonlinear evolution equations is the extended Tanh-function method.With the development of computer science,directly searching for exact soliton and soliton-like solutions of nonlinear evolution equations has become more and more attractive partly due to the availability of symbolic computation softwares like Maple or Mathematica which enable us to perform the complex and tedious computation on computers.We first show the exsisted Tanh-function method,and then improve this method to solve more NLEEs and obtain new exact solutions.