Exact Solutions Of Nonlinear Evolution Equations And Some Auxiliary Equations

The exact solutions of the nonlinear evolution equations is an important research area in soliton theory. It includes two routine. One of it is to construct new methods of seeking exact solutions of nonlinear equations.Another is to use the previously established method to give some new exact solution of the nonlinear evolution equation. Due to the complexity of the nonlinear evolution equation, it seems to be very difficult to establish a unified method for solving nonlinear equations. However, it is important to explore and find out special methods of solving nonlinear evolution equations. It is also possible to find new methods to solve some special nonlinear evolution equations according to their individual properties and some new solutions of these nonlinear evolution equations may be obtained by the established methods. These new solutions or special solutions may beused to explain the new physical phenomena.Therefore, constructing new methods to solve nonlinear evolution equations or finding new solutions of nonlinear evolution equations by the established method is very important to study nonlinear problems. The previously established methods are great interest to physicists and engineers because these methods can be directly used to solve nonlinear problems and explain physical phenomena in a very short time. In this sense, the study of the methods for solving nonlinear evolution equations play an important role to improve theory the theory of nonlinear equations and helpful to solve practical problems.This paper based on the research work of many experts and scholars studies some nonlinear evolution equations by using exp(-?(?))- expansion method, G'/G- expansion method and generalized auxiliary equation method and tries to construct their exact solutions. This paper includes four parts which is arranged as follows:The first chapter owes to brief introduction of the research work of the soliton theory , the generalized auxiliary equation method and the main work of this paper.The second chapter will briefly introduce the exp(-?(?)) - expansion method and apply it to solve the (2+1)-dependent long-wave equation, the generalized variable coefficients KdV-mKdV equation and the variable coefficient (2+1)-dimensional Broer-Kaup equation. As results, some new singular traveling wave solutions of these equations are obtained. Compares our results with the previously exact solutions given by different methods ,we find that those solutions are only special cases of the solutions given in this paper.In the third chapter, we shall introduce the G'/G-expansion method and use it to obtain the general solutions involving free parameters of the constant coefficients Newell equation, the variable coefficients Novikov-Ves elov equation and the discrete complex cubic - quaternary Ginzburg-Landau equation,etc. It is shown that the G'/G-expansion method can be used to solve the variable coefficient equations, complex equations and discrete equations and other types of nonlinear equations. The G'/G-expansion method is simple, straightforward, and can give more abundant exact solution of nonlinear equations due to its free parameters.In the fourth chapter we shall introduce and analyze the generalized auxiliary equation method which is used to obtain abundant exact soliton solutions of the (2 + 1) dimensional Calogero-Bogoyavlenskii-Schiff equation and the variable coefficient combination KdV equation.