The Discussion On The History Evolution Of The Auxiliary Equation Method For Solving Nonlinear Evolution Equations

In 1834, J. Scott Russell observed natural phenomenon of solitary waves. In 1895, D.J.Korteweg and G.de Vries governed mathematical model for descriptirrg phenomenon of Rus-sell solitary wave in KdV equation when they investigated shallow water wave in one dimensional, and confirmed the existence of solitary wave solutions in theory. In 1955, Enrico Fermi, John Pasta and Stan Ulam presented FPU problem which proposed the first experimental foundation for discoverying soliton.In 1965, N.J.Zabusky and M.D.Kruskal discovered the length between speings in FPU problem satisfying KdV equation and reappeared the nature of the particles as interaction of solitary wave by computer simulation. Because of this, Zabusky and Kruskal named these special waves as solitons, which testified stabilization of solitary wave solutions.In the rencent fifty years, with the development of the computer, optical soliton, Davydov soliton and inner solitary wave were discovered in the nonlinear optics, biology and oceanol-ogy, respectively. And smooth soliton solutions, peak soliton solutions and compact soliton solutions, etc in several fields are investigated, such as condense state physics, laser physics, superconductivity physics, economics, population problem, medical science and so on.Now we would like to extend the two contents of the study of solitary theory as follows(1) Constructing solving method of systems. The systematic method for constructing and developing solving nonlinear equations was presented, which include nonlinear partial differential equations, nonlinear ordinary differential equations, nonlinear integrable differential equations and nonlinear difference differential equations. Many powerful solving methods of nonlinear evolution equations are introduced, but a few common methods are proposed.(2) Explaining the property of solutions. One of our aim is to study and explain many more algebraic and geometric properties of integrable equations, where integrable equations are nonlinear equations which can transformed into linear equations. Here will give three aspects about properties of solutions. For example, firstly, we will analyze and study qualitative prob-lems of nonlinear evolution equations when explicit exact solutions are difficult to be obtained. Secondly, the solutions are simulated and analyzed with the help of computation mathematical theory and computer. Finally, exact solutions of nonlinear evolution equations are obtained using trial function methods and constructive transformation methods, etc. Though the three research methods are different, the aims are to explain change rule of solutions.The history of mathematics is to study origin and development of mathematics concepts, mathematics method and mathematics thinking, and the relation between these with social politics, economics and general cultures. In 1974, Wu Wentsun began to study the history of Chinese mathematics. Under "Make the Past to the Present" principle, he applied the method for combing anti-Whig with the comparison of Chinese and Western mathematics to study the Chinese classical mathematics and obtained the two characteristics of constructively and mechanization about Chinese mathematics. On this basis with computer, the world-famous "Wu elimination method" was presented. Therefore, the work of Wu Wentsun is a model of make the past to the present, and new methodology Wu proposed is the guidance and enlightening for the history and the research of mathematics.The construction of exact solutions to nonlinear evolution equations is one of important researches in the solitary theory. The auxiliary method and trial function method have played significance on constructing exact solutions of nonlinear evolution equations, and much more work has been done. In the dissertation, according to Wu elimination method and applying new methodology to study references on the auxiliary equation method and the trial function method before 2009, we have searched for the two characteristics of constructively and mechanization of two methods. Hence, in chapter 4, we will summarize the two characteristics of constructively and mechanization about, the trial function method and propose the new trial function method to seek new exact solutions of nonlinear continuous(or discrete) evolution equations.In chapter 5, based on the references, ideological foundation and sources of Riccati equation and other auxiliary equation methods, we sum up four applicable procedures to reflect the two characteristics of constructively and mechanization about auxiliary equation methods. Accord-ing to this, developing the characteristics of auxiliary equation method, the auxiliary equation methods of triangular function type and hyperbolic function type,etc. have been presented to construct exact solutions of nonlinear evolution equations.(1) There is constructive for becoming nonlinear envolution equations into nonlinear ordi-nary differential equations.(2) There is constructive for formal solutions of auxiliary equations and nonlinear ordinary differential equations.(3) There is mechanization for solving the sets of nonlinear equations.(4) There is mechanization for illustrating the solutions of nonlinear evolution equations.In theory, nonlinear envolution equations exist infinite solutions. However, the auxiliary equation methods in many Master's thesises, doctoral dissertations and references can obtain fi-nite exact solutions. Therefore, in this dissertation, to seek new infinite sequence exact solutions to nonlinear evolution equations, studying highly references and summing up the characteris-tics of the auxiliary equation method, the auto-Backlund transformation, the quasi-Backlund transformation and the formula of nonlinear superposition of the solutions of Riccati equation, the first kind of elliptic equation and the second kind of elliptic equation,etc. are presented to construct new infinite sequence exact solutions of continuous(or discrete) nonlinear envolution equations with variable coefficients(or constant coefficients) as follows.Firstly, infinite sequences exact solutions of single function. The infinite sequences exact solutions are constructed by Jacobi elliptic function, hyperbolic function, trianglar function and rational function. respectively, which include infinite sequences smooth solitary wave solutions, infinite sequences peak solitary wave solutions and infinite sequences compact soliton solutions. In this dissertation, beside K(m,n) equation, Degasperis-Procesi equation and CH equation, infinite sequences peak solitary wave solutions and infinite sequences compact soliton solutions are discovered in other nonlinear evolution equations.Secondly. infinite sequences exact solutions of composite function.The infinite sequences exact solutions are composited by Jacobi elliptic function, hyperbolic function, trianglar func- tion and rational function in several forms, where include infinite sequences exact solutions are combined by smooth solitary wave solutions, peak solitary wave solutions and compact soliton solutions.