A (G′/G)-expansion Method Based On A New Nonlinear Auxiliary Equation

It is an important and old research work for obtaining the exact solution of differential equations. The explicit solution, especially traveling wave solution, can describe many physical phenomena well, such as oscillation, propagation wave etc. Until now many important equations can't still obtain the exact solution since the complexity of nonlinear equations, so to seek new method or to extend existed method becomes valuable work.Firstly, we will put forward a new (?)-expansion method. In the second chapter, we explains in detail the new (?)-expansion method, and apply this method into a series of nonlinear evolution equations for the first time, and get rich exact solutions. In the third chapter, new (?)-expansion method are promoted in the further. Applied the promoted method to the (3+1)-dimensional potential-YTSF equation, we obtained rich families exact solutions.Secondly, we try to find a general method to solve nonlinear evolution equations. In the forth chapter, we propose a unified form method for the first time. A series of nonlinear evolution equations (group) are transformed uniformly into Aφ" + Bφ+ Cφ~2 + Dφ~3 = 0 form. Through solving the ordinary differential equation to solve the evolution equations.