Soliton Solutions Of Nonlinear Partial Differential Equations

Nonlinear dynamics is one of the important branches in nonlinear science, whereas study on how to solve nonlinear partial differential equations is one of the main contents in nonlinear dynamics. As a leading subject and hot interest in nonlinear science, study on the solution method of nonlinear partial differential equations has become more and more challenging. At present, although a number of methods are proposed and developed to look for the exact solutions of nonlinear partial differential equations, unfortunately, not all these approaches are universally applicable for solving all kinds of nonlinear partial differential equations directly. As a consequence, it is still a very significant task to go on searching for various powerful and efficient approaches to solve nonlinear partial differential equations, In this thesis, based on a complete summary and examination of the main methods for solving nonlinear partial differential equations, a few new approaches are proposed to seek the exact solutions of nonlinear partial differential equations. By making use of these approached proposed by us, a variety of exact solutions to many physically significant nonlinear partial differential equations are easily presented. Among these solutions some are in very good agreement with those obtained in other literatures, and some are new ones which can not be found in the existing literatures to the best of our knowledge. The studies are of more profoundly theoretical significance and important application value.The dissertation consists of five chapters. In chapter one, the history of the development for nonlinear partial differential equations is looked back on, and the nonlinear partial differential equation's future is predicted. In chapter two, the main methods for solving of nonlinear partial differential equations are summarized, and the primary contents of this dissertation are reported as well as. In chapter three, a new approach called the simple equation method is proposed to solve nonlinear partial differential equations, and by using it, the SK equation is solved. In chapter four, Backlund transformation is used to discuss Burgurs equation, and simple soliton solution, two soliton solution and three soliton solution of the equation are obtained. Applying the results, we study the phenomenon of the soliton fission and fusion. In chapter four ,one of the main methods for solving nonlinear partial differential equations-the first integal method is used to solve nonlinear partial differential diffused equations, and some new exact solutions are obtained. At last, the work of this dissertation is summarized, and further work in the future is proposed.