Construction Of Some Nonlinear Evolution Equations' Exact Solutions

This dissertation mainly studies and analyses the following aspects:Finding exact solutions of some nonlinear evolution equation(s) arrising from physics. The main problems under consideration is how to seek efficient transformations that can reduce the equation(s) to be solved to one(s) which is (are) simpler and easily solved. By solving the latter, the exact solutions of the original equation(s) can be then obtained.The main results derived are as follows:(1). In chapter 2, the general principle to solve nonlinear partial equations is introduced. Some illustrative examples are presented to show how to use the principle and the application range.(2). In chapter 3, we introduced the further extended Projective Riccati method, applied it to (2+1)-Bugers equation and (2+1)-dimensional breaking soliton equation, and obtained a lot of soliton and soliton-like solutions. By introducing a new variable, the (2+l)-dimensional breaking soliton equation is simplified to be a sigle one. The latter one is not only with a low dimension, but also a low order. This makes it much simpler to use the efficient extended projective Riccati equation method used in recent years in the literature. In particular, when seeking soliton-like solutions of this kind of nonlinear evolution equations, the concise degree is much notable, even more simpler than the method existing to find travelling wave solutions.The dissertation is organized as follows. In chapter 1 the development of solving methods for finding exact solutions of nonlinear evolution equations is introduced briefly. In chapter 2, under the guidance of the theory of AC=BD presented by professor Zhang Hongqing in 1978, we introduce general principle of seeking solutions for differential equation and its application to searching for exact solutions of nonlinear evolution equations. The two aspects mentioned above are discussed in details in chapter 3.