Several Of Exact Solutions Of Nonlinear Evolution Equations

With the rapid development of computer technology and the linear theory being perfect increasingly, the nonlinear science, which is extensively found in physics, chemistry, biotechnology, geology, economy, finance and other areas, has played more and more important role in engineering and natural science. There-fore, the practical problem is characterized by the mathematical language, and the further analysis and research are in progress with the use of mathemati-cal concepts, methods and theories, thus the practical problem is characterized from the qualitative or quantitative term and the guidance is provided to solve the practical problem. In fact, this course is building the mathematical mod-els, analyzing and settling them. The analysis of these models can be largely attributed to solve the model equations, including linear or nonlinear ordinary differential equations, partial differential equations, functional equations and dif-ferential equations, etc.. So, how to solve these equations becomes the emphasis and difficulty of the nonlinear science.Because of the complexity of the partial differential equations, there is still not the unified method which is fit for solving the exact solutions of the whole partial differential equations, and the exact solutions of a large number of im-portant equations can not be obtained. Therefore, it remains one of the very important subjects of the nonlinear science to look for new and effective meth-ods to deal with the nonlinear partial differential equations.In this paper, we utilize the double-function method and the extended tanh-function method to study several nonlinear evolution equations and get some new exact solutions of them. These two methods both translate partial differential equations into the simple ordinary differential equations by the traveling wave transformation, then these ordinary differential equations can be solved to obtain the traveling wave solutions of the given partial differential equations. In chapter one, we introduce several methods to obtain the exact solutions of nonlinear partial differential equations. In chapter two, we use double-function method to obtain the new exact solutions of a kind of nonlinear evolution equation and exact solutions of some equations, which are the special cases of the nonlinear evolution equation, are obtained. In chapter three, we utilize the extended tanh-function method to solve the famous Peregrine-Benjamin-Bona-Mahoney equation and (2+1)-dimensional dispersive long-wave equation respectively and get some more exact solutions of them. Finally, a summation is made to generalize the work in this paper. Suggestions are proposed for further research and improvement.