The Research Of Solving The Nonlinear Evolution Equations In Soliton Theory

Soliton theory in the natural science research plays a very important part. The soliton theory of the nonlinear evolution equations is the important hot spot topic. Some nonlinear evolution equations with physical background have soliton properties. Therefore, the important significance in theory is to get solutions of the nonlinear evolution equations. At present, a number of methods are proposed to look for the exact solutions of the nonlinear evolution equations, for example, inverse scattering transformation method, function expansion method, deformation mapping method, mixing exponential method, bilinear method and Darboux transformation method. As for solving nonlinear evolution equations have not an unified way. As a consequence, it is still a task for further research and issue to go on searching for efficient approaches to solving nonlinear evolution equations.This dissertation is based on systematic research and on the existing technique of solving nonlinear evolution equations and the existing soliton theory. Some methods for constructing the exact traveling wave solutions of the nonlinear evolution equations are applied and improved, new exact solutions of several types have been obtained.This dissertation consists of three chapters.In Chapter 1, we introduce the historical background and the development of the soliton theory, integrability of the soliton, study development of nonlinear evolution equation and several commonly used methods for solving the nonlinear evolution equation. Then briefly describes the content and significance in this dissertation.In Chapter 2, based on the methods of solving nonlinear evolution equations, we firstly extend the existing tanh-coth method, the method is applied to the generalized Zakharov equation and we obtained a series of exact solutions. By using the second homogeneous balance method with a differential equation and Coupled projected Riccati equations, solving generalized Zakharov equation, extending the structure of the solution obtained of the original method. Secondly, by using the promotion of the Jacobi elliptic function method, we obtain the new elliptic function solutions of the (2+1) dimensional Konopelchenko-Dubrovsky equation. In the limit cases, these solutions degenerate to solition solutions and triangular function solutions. Finally, by using the improved (G'/G) expansion method, solving the (2 +1)-dimensional breaking soliton equation, we find abundant exact solutions. The solutions are expressed by hyperbolic functions, trigonometric functions and rational functions contained arbitrary parameters.