The Multi-order Exact Solutions And Periodic Solutions To The Nonlinear Evolution Equations

Recently, much effort has been spent on the construction of the exact solutions, especially the exact solitary wave solutions of the nonlinear evolution equations (NLEEs). Many new methods for obtaining the exact solutions of NLEEs are proposed, such as the homogeneous balance method, the trial function method, the tanh-function method, the sine-cosine method and Jacobi elliptic function expansion method, etc. In order to discuss the stability of these solutions, a perturbation must be add to these solutions, and the evolution of the perturbation should be analysed. This method, in fact, is to expand the solutions of the NLEEs to a power series of parameters, and further to obtain its every order exact solutions.In this paper, a kind of hypothesis of solution and perturbation method are used to obtain the multi-order and high-order exact solutions of NLEEs.Preface introduces the developmental status of solving the exact solutions of NLEEs.The part one introduces the multi-order exact solutions of NLEEs. By using a kind of hypothesis of solution, the exact solutions of three representative NLEEs are obtained. In paper [35], Liu et.al used Lamefunction and perturbation method obtained the one-order and two-order exact solutions of KdV equation, nonlinear Klein-Gorden equation and mKdV equation, while in this paper , Lame function is not used in the process of constructing the exact solutions of NLEEs. So the method in this paper expands the method proposed by Liu ed.al, can be applied to more equations.The part two discussed the high-order exact solutions of NLEEs. On the basis of the part one, with the help of Mathematica system, the high-order exact solutions of NLEEs are obtained, and the general expression of the high-order exact solutions are summarized.In the part three, the exact solutions of variable coefficient nonlinear evolution equations are obtained by using auxiliary equation method. The variable coefficient KdV equation is given as an example to introduce the application of this method, and some new elliptic function solutions are constructed.