Solving The Nonlinear Evolution Equations And The Darboux Transformation

The paper mainly discusses solving the nonlinear evolution equations, which is an important problem of the soliton theory. And some methods of constructing the nonlinear evolution equations are improved. The paper consists with three chapters.Chapter 1 is to introduce the history and development of the soliton theory, the history and development of the nonlinear evolution equations and to generalize the methods of constructing the nonlinear evolution equations. My main works are listed at last.Section 1 in chapter 2 is to make use of the extended F-expansion method,and improve the form of the solutions for the Zakharov equations to construct more general solutions, and thus get several sets of new exact solutions, including the trigonometric function solutions, the hyperbolic function solutions, the Jacobi elliptic function solutions .Section 2 is under the guidance of the improvement and generalization of function expansion method, several methods are combined. Take example of the variable coefficient Fisher equation, and suppose the form of the solutionsSome new exact solutions are obtained .Chapter 3 mainly makes use of the Darboux transformation to solve the 2+1 equations. The Darboux transformation is an important method to solve the nonlinear evolution equations, which needs technique. At first, the chapter introduces the Darboux transformation chiefly, then supposes the two the Darboux transformations for the 2+1 Levi equations, and some new soliton solutions are obtained. Thereinto,Q=(?),q_k=(?),andαis a function about x,t.