Exact Solutions To Some Important Nonlinear Evolutionary Equations

In this paper, along with the orbit of the development of nonlinear science, we reviewed the theory of exact solutions about nonlinear science entirely, and further research the method used to construct exact solutions which mastered by the people till now, analyze and summarize the merit and demerit of each method. Ameliorate to the Riccati equations method, simplified, enriched and expanded the result we have known. Which has active meaning for us to find new soliton solutions, research the long time dynamic behavior and structure of solitons .at the same time, classified the structure of solitons, introduce the research of the theory related, including the Localized excitations to several kinds of important high-dimension nonlinear system.In the first chapter, we study the history, current situation and future of our research work, at the same time introduce the main work of this paper.In the second chapter, we introduce some primary conception and denotation relate to this paper, give the definition and befallen mechanism of solitons, discuss the similarities and differences between soliton with solitary wave, classified the solitons which we have known according to the dimension of the space, Make out some explain necessary to the exact solutions, approximately solutions and similar solutions which tend to be confused. At the same time, we present the definition and distinguish of solitary equations and integrable system.In the third chapter, we introduce the methods for finding exact solutions which people have mastered till now, analyze the merit and demerit of each method, and prospect the direction of development for the future. In the fourth chapter, we study the exact solutions to several kinds of important nonlinear evolution equations by using several different methods, we obtain a lot of new significative solutions except many solutions which we have known.In the fifth chapter, we Ameliorate to the traditional Riccati equations method, and use this method to research the exact solutions of several kinds of nonlinear evolution equations with constant coefficients or variable coefficients.In the sixth chapter, we study the Localized excitations to several kinds of important nonlinear evolution equations.In the seventh chapter, we prospect the future of exact solutions' theory of nonlinear evolution equations.