Study On The Theories And Algorithmes For Constructing Exact Solutions Of Nonlinear Evolution Equations

Nonlinear dynamics is one of the important branches in nonlinear science, and its main research objects are bifurcation, chaos, fractal and solitons, etc.whereas study on how to solve nonlinear evolution equations is one of the main contents in soliton theory. With the development of computer technology, especially the emergence of computer symbol computation softwares, study on the theories and algorithmes for constructing exact solutions of nonlinear evolution equations has become a leading subject and new interest in nonliner science. At present, although a number of algorithmes are proposed and developed to construct exact solutions of nonlinear evolution equations, unfortunately, not all these approaches are universally applicable for solving all kinds of nonlinear evolution equations directly. As a consequence, it is still a very significant task to go on searching for various powerful and efficient approaches to solve nonlinear evolution equations. In this thesis, based on a complete summary and examination of the main methods for constructing exact of nonlinear evolution equations, a systematic investigation into the fundamental theories and algorithmes of looking for exact solutions of nonlinear evolution equations has been studied, and a few new approaches are proposed and developed to seek exact solutions of nonlinear evolution equations. By making use of these approaches proposed, a variety of exact solutions to many physically significant nonlinear evolution equations are obtained. Among them, some are in agreement with those obtained in some literatures, others are new ones which can not be found in the existing literatures to the best of our knowledge. Our studies enrich and develop the theories and algorithmes of constructing exact solutions of nonlinear evolution equations, and have more profound theoretical significance and important application value.This dissertation consists of seven chapters. In chapter one, the main theories and algorithmes of constructing exact solutions of nonlinear evolution equations are summarized, and the source of the subject, the research purpose, the primary contents and innovations of this dissertation are reported as well. In chapter two, several important nonlinear evolution equations in mechanics are derived, and it indicate that the nonlinear evolution equations involved in our study have a mechanics or physics background. In chapter three, We extend Fan's algebraic method to its auxiliary ordinary differential equation's degree more than 4 by using a complete discrimination system for polynomials, and apply this method to construct a series of traveling wave solutions for KdV equation and the variant Boussinesq equations including rational solutions, solitary wave solutions, triangular periodic solutions, Jacobi periodic wave solutions and implicit function solutions. In chapter four, an auxiliary ordinary differential equation method and a direct method are presented by introducing an auxiliary ordinary differential equation whose some new exact solutions are obtained. By using the auxiliary equation method the Combined KdV-mKdV equation, (2+1)-dimensional Broer-Kaup-Kupershmidt system and two kinds of KdV equation with variable coefficients are investigated and abundant exact traveling wave solutions are obtained that include solitary wave solutions and triangular periodic wave solutions. By making use of the direct method, the same kinds of solutions of (1+1)-dimensional Klein-Gordon equation, (3+1)-dimensional Kadomtsev Petviashvili equation, the (2+1)-dimensional dispersive long wave equations and the Klein- Gordon-Zakharov equations are obtained. In chapter five, some new exact solutions of the generalized Lienard equation are obtained, and the solutions of the equation are applied to construct directly the exact solutions of the generalized one-dimensional Klein-Gordon equation, the generalized Ablowitz equation and the generalized Gerdjikov-Ivanov equation. Then based on an auxiliary ordinary differential equation with nonlinear terms of any order, a new approach called the auxiliary equation with nonlinear terms of any order method is proposed to construct the exact solutions to the generalized Zakharov equations and the generalized Benjamin-Bona-Mahony equation. In chapter six, the sinh-Gordon equation expansion method is further extended by generalizing the sinh-Gordon equation and constructing new ansatz solution of the considered equation. As its application, the (2+1)-dimensional Konopelchenko–Dubrovsky equation, the KdV-mKdV equation, the double sine-Gordon equation and the BBM equation are investigated and abundant exact travelling wave solutions are explicitly obtained including Jacobi elliptic doubly periodic function solutions, solitary wave solutions and trigonometric function solutions. In chapter seven, a new discrete coupled Riccati equations expansion algorithm is presented to construct the exact solutions of nonlinear differential-difference equations by introducing the coupled Riccati equations. As an example, we apply this algorithm to the general lattice equation, the relativistic Toda lattice equations and the (2+1)dimensional Toda lattice equation. As a result, some kink solitary wave solutions and complexiton solutions of them are obtained with the help of symbolic system Mathematica. Finally, the summary of this dissertation and the prospect of study on constructing exact solutions to nonlinear evolution equations are given.