Studies On The Integrability And Exact Solutions Of Three Nonlinear Systems

Soliton theory plays an important role in the nonlinear science.It is known that the nonlinear partial differential equation usually arises from many fields of science such as information science,nonlinear optics,chemistry and oceanography,etc.If the exact solution of a special equation can be obtained,it will be helpful to clarify the movement of the objects with the nonlinear terms and it can also interpret various nonlinear phenomena accurately as well as discovering new laws of natural phenomena.In recent years,with the development of computer technology,how to find the exact solutions of nonlinear evolution equation is becoming an active field and many new methods for constructing exact solutions are proposed.This dissertation is devoted to analyze some methods which are effective to obtain exact solutions of nonlinear evolution equations and we will construct many interaction solutions of three nonlinear systems by the means of Painleve analysis,consistent Riccati expansion and consistent tanh expansion method.The main work is carried out in the following aspects:Firstly,we briefly introduce the history and research background of soliton theory.In addition,the general steps of Painlevé analysis,CRE method,CTE method are proposed and the significance of the thesis is also given out.Secondly,we study the Painlevé integrability of the coupled Higgs system by the Kruskal's simplification of WTC method,and new exact solutions are constructed by the Painlevé standard and nonstandard truncation expansion.Then we use a special form of Painlevé truncated expansion to study the coupled Higgs system and two types of elliptic function solutions are obtained by using M?bious transformation.Thirdly,a consistent Riccati expansion(CRE)method is developed for a special KS equation,and we prove the CRE integrability and obtain the soliton-cnoidal wave interaction solution of the special KS equation.Besides we prove the general KS equation is non-CRE solvable finally.Finally,a consistent tanh expansion(CTE)method is developed for the combined Kd V-m Kd V equation,and the equation is proved to be CTE solvable.Many exact interaction solutions such as soliton-cnoidal wave solution,soliton-periodic wave solution for the combined Kd V-m Kd V equation are given out analytically and graphically.Summary and discussions are given in last section.