Several Methods Of The Investgating Exact Solutions For Nonlinear Partial Differential Equations

In the 21 st century, the mainstream direction of scientific research and technological development is nonlinear science. How to construct the exact solutions of nonlinear evolution equations is an important branch of nonlinear science research. At present, the theory and algorithms for constructing exact solutions of nonlinear evolution equations have been proposed and developed, but the method for constructing the exact solutions of nonlinear evolution equations is not universally applicable. Therefore, it is still a valuable research work to continuely look for some e?ective solving method.With the development of computer symbolic system, many experts and scholars have done a lot of work in the exact solutions to nonlinear evolution equations, constructing many kinds of e?ective methods. This paper is on the basis of previous studies, focusing on the simple equation method and programming this method successfully. In addition,it is based on Hirota bilinear method systematically that the paper has researched on how to construct Rimann theta function periodic wave solutions of(1+1)- dimensional nonlinear evolution equations. In this paper, a simple equation method is applied to some important nonlinear mathematical physical model from a low dimension to high-dimensional. We successfully obtain the rich travelling wave solutions and analysis the state of the exact solutions by images. The Rimann theta function periodic wave solutions of 9 order Kd V equation are derived. The di?erence between the two methods is that the former theory is simple but the large amount of calculation, the Mathematica mathematical software is also very easy to operate. The advantage of the latter is that it only relies on the existence of Hirota bilinear forms. Moreover, all parameters appearing in Riemann matrices are completely arbitrary,whereas algebro-geometric solutions involve specific Riemann constants, which are usually di?cult to compute. The chapters and contents are as follows:In the first chapter, we introduce the research background and the current state of development of the research contents in this paper, and briefly explain the main work of this paper.In the second chapter, the simplest equation method is employed to construct the bell shaped soliton solutions of the(2+1)- dimensional KP equation, and the solutions were analyzed.In the third chapter, the key processes of the simplest equation method are summarized, and this method is extended to the three class of nonlinear mathematical physics model which contains Benjamin-Bona-Mahoney equation,(2+1)-dimensional Boussinesq equation,(3+1)-dimensional YTSF equation, obtaining traveling wave solutions. In the fourth chapter, the research status and significance of Rimann theta periodic wave method are given, and the method is extended to a high dimensional integrable models of 9-order Kd V equation. Then the one-periodic and two-periodic wave solutions of the equation are obtained. At last, the propagation behavior of periodic wave solutions are clearly described, and the relationship between the one-periodic wave solutions and the two-periodic wave solutions are established.The conclusions and future researches are prospected in the end.