A Number Of Research Of Solving The Nonlinear Equations

This paper studies the soliton theory in several important soliton equations to solve the problem by using the homogeneous balance method and bilinear method,and the important in the practical application of soliton equation. This article also introduces the similarity transformation and bilinear method to solve KdV equation with variable coefficients. The main contribution of the paper is listed as follows:The first chapter mainly introduces the historical development of the soliton theory and research results obtained by one stage at that time, meanwhile, the subjects of domestic and foreign scholars’ achievements are also explained in detail. It also describes the development of soliton solution and soliton in fluid mechanics, plasma physics, optics, quantum theory and some other applications.In the second chapter, based on the principle of homogeneous balance, some new techniques are involved in the homogeneous balance method, and some special equations are illustrated. Including MKdV equation, Huxley equation, Burgers equation, Chaffee-Infante equation. The solution of these equations has certain practical significance to explain some physical phenomena.The third chapter is mainly divided into two aspects to expound the bilinear method. In this paper, we first introduce the principle of bilinear method and some properties of bilinear operator, and use the bilinear method to solve the differential equation of constant coefficient differential equation, and obtain the single soliton solution and double soliton solution of the equation. Secondly, we apply the self-similar transformation and Hirota method to the KdV equation with variable coefficients, we obtain a single soliton solution and a double soliton solution of the KdV equation with variable coefficients.