The Function Expansion Method And Exact Solutions Of The Nonlinear Partial Differential Equations

It is an important and old research subject for obtaining the exact solution of differential equations. The explicit solution, especially the traveling wave solution, can be used to describe many physical phenomena well, such as oscillation, propagation wave etc. Up to now the exact solutions for many important equations can not still be received since the complexity of the nonlinear equations. Thus seeking new method and extending existed method, both come to be an important and valuable work.In this paper, we generalize the Tanh-function method, exhibit the united formulae to solutions of constant coefficient equations and variable coefficient equations, by the generalized method we can get more exact solutions. Meanwhile it is convenience to be dealt with by the computer. In order to receive more solutions, we also modify the subsidiary equation to beφ'(ξ) =φ~2(ξ) + b(ξ), where b(ξ) is an arbitrary function ofξ. The arbitrariness of b(ξ) makes the solutions more flexible. Specially, the case when b(ξ) = kξ~n is discussed in this paper. And by applying the generalized method to KdV equation, (2+1) dimension KD equations and other nonlinear evolution equations, we obtain abundant exact solution families of those equations.Based on the homogenous balance method, by introducing the parameters we give a new method of solving nonlinear partial differential equation (NLPDE)—Method of parametric transformation. For some specific NLPDE we obtain the multi-soliton solutions, self-similar solutions and trun- cation series solutions associated with the GM (Gardner-Morikawa) transformation.