Exact Solutions And Analytical Approximate Solutions Of Nonlinear Wave System

Seeking the solution of nonlinear PDE, has opened a new frontier and hotspot research in the field of nonlinear science, which is quite challenging. Many efficient methods for finding exact solutions for nonlinear PDE have currently been proposed and developed by the scholars at home and abroad.. There is no universal to solve nonlinear PDE. It is extremely important and valuable to continue the study to explore some effective and feasible methods. This dissertation is based on systematic research and on the existing technique of solving nonlinear PDE. Ameliorating the Riccati equation method simplifies, enriches and helps develop the existing methods. This obtains not only existing results but also, meaningful new solutions as well. This sums up, amends and extends the traditional F-expansion method by employing the amended Riccati equation method, the Backlund transform method, and the F-expansion method. This will undoubtedly have a positive influence on the study of the new solitons' discovery, solitary wave equation's longtime dynamical behavior, and the solitons structure.However, we can just get part of the solutions due to the complexity of the partial differential equations themselves, before we can build a method to construct the general solutions of them. Besides there are just a few equations that can be exactly solved. Therefore, seeking the approximate solutions of these equations becomes a hot subject. This, accompanied by the development of computer science, makes it possible to seek analytical approximate solutions of nonlinear PDE by computer. In the frame of TTE, this work proves that the resolvent operator of compound KDV equation is computable. Therefore, one can give the analytical approximate solutions of the system's initial value's problems by using a computer. This work enriches and develops the contents of nonlinear PDE solution and has great theoretical sense and application values.This whole work is presented in 8 chapters. Firstly, an overview is given on the wave equation issue, its solution, along with the historical and present studies of computability theory. In chapter 2, we introduce the solitary wave, its definition and generation mechanism and classify these solitons that are known up to now. Chapter 3 presents the basic concepts related to this work and proposes several basic concepts and computability properties on computability space. Chapter 4 amends the traditional Riccati equation method and studies the exact solutions of nonlinear wave equations, by using the above method. Chapter 5 uses Backhand transform, N+1 soliton solutions of compound KdV-Burgers equation with variant coefficients are obtained. Chapter 6 gives the solitary wave solutions of compound KdV-Burgers equation with variant coefficients and N+l dimensional Sine-Gordon equation. In Chapter 7, we prove the resolvent operator of compound KDV equation is computable via the operator method, contraction mapping principle and TTE theory. Finally, we summarize the whole research and give an outlook in the direction of this research.