Study Of The Exact Travelling Wave Solutions Of Some Nonlinear Evolutionary Equations

Nonlinear phenomenon is the most common phenomenon in nature, and it is the essence of nature. When nonlinear systems were proposed and researched, it promoted fusion of different disciplines. A large number of new disciplines emerged, and gradually gave birth to the nonlinear science to systematically study the complexity of the phenomenon. Nonlinear science includes the research of soliton theory, chaos theory and fractal theory and theory of dissipative structures, etc., and the extensive application of these theories in other related disciplines. Solitary wave theory and the application is a hot topic in nonlinear science.The related properties of soliton play great importance in revealing the wave propagation, in the accurate and scientific explanation of natural phenomenon, and in the related application of engineering techniques. In order to describe these, the nonlinear evolutionary equation (or equations) was introduced, and therefore there are both theoretical and practical meanings both in quantitative research and qualitative research. The research of amount of nonlinear problem comes to the research of the nonlinear evolutionary equation (or equations).However, it is more sophisticated to solve the nonlinear equation than to solve linear one, and generally there isn't a unified approach to deal with the former. Thus, it shows a very important theoretical and practical value for solving nonlinear partial differential equation (or equations).Based on this, in the summary of the major existing methods for solving nonlinear evolution equations, the paper applies several methods to some typical equations, such as mBBM equation, MCH equation, Klein-Gardner equation, combined KdV-mKdV equation, generalized BBM equations. From the application, the paper got some old solutions and some new ones.The full-text is divided into seven chapters. The first chapter describes the research background, progress and nowadays works of nonlinear wave equation.The second chapter introduces several important methods for looking for exact solutions of nonlinear wave equation, and briefly describes the basic concepts and principles in this paper for solving nonlinear wave equation associated with this article.In the third chapter, by means of the first integral method, I analyzed some equations, such as mBBM equation, simplified form of MCH equation, Klein-Gardner equation, and combined KdV-mKdV equation. By a comprehensive analysis, I find some exact solutions, combined with a simple direct integration and the expansion method of Jacobi elliptic sine function.In the next chapter, I apply (G'/G) expansion method to discuss the simplified form of MCH equation, Klein-Gardner equation, combined KdV-mKdV equation, and get many solutions with the forms of hyperbolic function, trigonometric function, which enrich the discussion of these equations. The increasingly rich application of the method is still suitable for some important discrete solitary wave equation.In the fifth chapter, inspired by Wazwaz's original work with well-known Hirota method, I apply a simple way to effectively search for the solutions of the (2 +1)-dimensional Zakharov-Kuznetsov equation (abbreviated as ZK (m, n, k)), breaking soliton equation, Potential Kadomtsev-Petviashvili (PKP) equation, and a fifth-order dispersion equation, and includes multiple soliton solutions and singular soliton solutions. Still I attempt to utilize the homoclinic test method to discuss the extension work of the Hirota method.In the sixth chapter, I am enlightened by the work of Kuru's decomposition theory of second order differential operator, and discuss several classes of generalized BBM equations which were researched by Wazwaz and two modified Boussinesq equations. After traveling wave transformation, with the Weierstrass function form, the paper obtains some types of traveling wave solutions, among which are periodic solutions and hyperbolic function solutions, and most of the solution has not been found in the literature.Finally, there are some of the conclusions of the various methods. I make some summary and forecast. Based on these, I would make future attempts.