The Research Of Complexion Solutions Of Some Kinds Of Nonlinear Evolution Equations And The Property Of The Solutions

Soliton theory occupies a very important place in the field of nonlinear science. A large number of nonlinear evolution equations have been found in the process of research. In order to get a further understanding of the practical significance of the nonlinear evolution equations, the most important step is to obtain a large number of new solutions.So far, there is not a unified method to solve plenty of nonlinear evolution equations due to the complex nature of them. In all of the methods, auxiliary equation method is more direct and more effective. In this paper, we mainly present a method combined with function transformation and auxiliary equation, and construct some new composite solutions of several kinds of variable coefficient(constant coefficient) nonlinear evolution equation(equations set) by employing the symbolic computation system Mathematica.These solutions include the Airy function, Jacobi elliptic function, hyperbolic function, trigonometric function and rational function combination.In the first chapter, the background of soliton theory have been described briefly. And introduce several solving methods of nonlinear evolution equations and the main task of this paper.In the second chapter, variable coefficient sine - Gordon equation can be turned into two-dimensional linear wave equation to solve through function transformation. Then, the new solutions of variable coefficient sine-Gordon equation can be constructed by using the solution of wave equation. Besides,some properties of the solutions are studied through their images.In the third chapter, mKdV equation, Sharma-Tasso-Olver(STO) equation and mZK equation have been turned into Airy function to solve through function transformation. Based on this, the Airy function solutions of nonlinear evolution equations such as mKdV equation are obtained with the help of Airy function and its solutions, and some properties of the solutions are researched with their images.The fourth chapter mainly focuses on the following there tasks.1. Based on the known solutions of the second kind of elliptic equation and nonlinear superposition formula of the solutions, the new infinite sequence composite solutions of the coupled KdV equations is constructed. The new solutions include Jacobi elliptic function, hyperbolic function and triangular function. And by the image of the solutions, some properties of the solutions are researched.2. By using the method combined with function transformation and second order homogeneous linear ordinary differential equations (or Riccati equation),the new composite solutions of variable coefficient (3+1)-dimensional breaking soliton equation are constructed. And by the image of the solutions,some properties of the solutions are studied.3. Through the function transformation, variant Boussinesq equation set can be turned into first order homogeneous linear differential equation and the solution of second order homogeneous linear differential equations to solve.On such basis, the new infinite sequence composite solutions of the variant Boussinesq equation set are constructed. And some properties of the solutions are reasched.