Some Investigation On The Methods For Finding Exact Solutions Of Nonlinear Evolution Equations

This dissertation mainly studies 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, we introduce the general principle to solve nonlinear partial differential equations. Some illustrative examples are presented to show how to use the principle and the application range.(2). In chapter 3, homogeneous balance method is used to deal with some problems of finding solutions of higher dimensional nonlinear evolution equation(s). By improving some key procedure of the method, the original equation(s) is (are) reduced to a linear equation(s), hence a great number of exact solutions for this (these) are obtained.(3). By introducing a new variable, the (2+l)-dimensional breaking soliton equation is simplified to be a sigle one, or using Backlund transformation to change a class of (2+l)-dimensional coupled equations to a (1+1)-dimensional nonlinear equation with only an unknow variable. 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 tanh-function method and the 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. For more details, please consult chapter 4-5, respectively.(4). Chapter 6 further investigates the variable separation method to solve nonlinear evolution equation(s) presented by professor Lou senyue. By usingBacklund transformation and Cole-Hopf transformation, some nonlinear equation(s) can be reduced to a linear partial equation including one or two arbitrary functions of two varable (x,t) or (y,t), respectively. Simlar to professor Lou's variable separation method, a new type of variable separation solutions are found. At the same time, some same type solutions obtained by professor Lou are also derived, but the method used here is more simpler.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 four aspects mentioned above are discussed in details in chapter 3-6, respectively.