Breather Solutions Some Nonlinear Partial Equation

The investigation of the exact solutions of nonlinear evolution equations play an important role in the study of nonlinear physical phenomena. For example, the wave phenomena observed in fluid dynamics, plasma and elastic media are often modeled by the bell-shaped sech solutions and the kink-shaped tanh solutions. In this article, we firstly introduce what are exact solutions, why study them? Then, we give out some methods to find out the exact solutions of nonlinear differential equation. By the way, we describe some physics phenomenons, these interest phenomenons urges us to find new physical phenomenon. At last, on the basis of the Hirota direct method, Professor Dai obtain new solution by extend test functions called extend homoclinic test function method. This method is concerned as a general method to solve nonlinear partial differential equatio, whose solution is explained as what will happen if soliton come up against periodic solution. As a result, we choice two equations, one is Schr?dinger-Boussinesq equation which is a representation of complex equation, the other one is Sinh-gordon equation, a real equation, to test our method. Based on their bilinear form, we pick out purposely two kind special function to find out homoclinic wave and breather solutions, at the same time, we found breather solutions have some homoclinic quality of Schr(o|¨)dinger-Boussinesq equation. For Sinh-gordon equation, we found its periodic solition solution by the help of our method.