The Sub-equation Method And The Exact Solutions For Some Nonlinear Evolution Equations

With the development of science and technology, nonlinear problems are presented in many fields, and some problems are described by the nonlinear evolution equations. In order to understand the physical meanings of the nonlinear evolution equations, it is one of the most important step that exact solutions of the nonlinear evolution equations are derived. So far, there is not a unified method to obtain the exact solutions of the nonlinear evolution equation because of the complexities of nonlinear evolution equations. Therefore, it is very valuable to find the exact solution of the nonlinear evolution equations in the theory and the application.With the efforts of many mathematicians and physicists, many methods to derive exact solutions have now been developed, such as the inverse scattering method, Darboux transformation method, Backlund transformation method, bilinear method, Lie group method, homogeneous balance method, dressing method, auxiliary equation method and so on. In these methods, the auxiliary equation method is paid attention due to the direct, concise and effective.Based on the auxiliary equation method, the topics on finding the exact solutions of nonlinear evolution equations are studied in this paper:(1) the g Kd V-qRLW equation, gKawahara equation, the generalized symmetric regularized long wave equation, gZakharov equation and Klein-Gordon-Zakharov equations with a highnonlinear term of arbitrary order are solved by using the auxiliary equation with single high-order power term and high order auxiliary equation with two high-order power terms respectively.(2) F/ G- expansion method are extended, and the mKdV equation with variable coefficients, KdV equation with variable coefficients and the(3+1) dimensional cubic-quintic Gross-Pitaevskii equations are studied, and exact solutions are obtained using the extended F/ G- expansion method.