Study Of The Exact Solution Of Nonlinear Wave Equations And Branch Issues

Nonlinear wave equations are important mathematical models for describing natural phenomena and are one of the forefront topics in the studies of nonlinear mathematical physics, especially in the studies of soliton theory. The research on finding explicit and exact solutions of nonlinear wave equations and on analyzing the qualitative behavior of solutions of nonlinear wave equations can help us understand the motion laws of the nonlinear systems under the nonlinear interactions, explain the corresponding natural phenomena reasonably, describe the essential properties of the nonlinear systems more deeply, and promote greatly the development of engineering technology and related subjects such as physics, mechanics and applied mathematics.In this dissertation, the exact travelling wave solutions, the bifurcations and dynamical behavior of travelling wave solutions of the nonlinear wave equations are investigated from the viewpoint of bifurcation theory of dynamical systems. Firstly, based on the main methods for finding exact solutions of nonlinear wave equations, the bifurcation theory and method of dynamical systems are used to improve a sub-equation method which is an effective method of finding exact solutions of nonlinear partial differential equations. By making use of the improved sub-equation method, a variety of new exact solutions to many physically significant nonlinear equations of mathematical physics are obtained. The research enriches and develops the approaches to exact solutions of differential equations. On the other hand, by using bifurcation theory of dynamical systems and singular perturbation theory, the qualitative behavior of travelling wave solutions to the several families of nonlinear wave equations from physical background are studied, and the rich dynamical properties of these nonlinear models are shown. Moreover the dynamical properties of singular homoclinic orbits are considered and first obtained, and the reason why the complex travelling wave solutions appear is successfully analyzed and explained. The results obtained develop the three-step method, an approach of dynamical systems proposed by professor Jibin Li for studying singular nonlinear wave equations.The major works of this dissertation mainly are as follows.In Chapter 1, the historical background, research developments, main methods and achievements of nonlinear wave equations are summarized. The discovery, correspond- ing research approaches and recent advance of non-smooth waves are introduced. The relation between nonlinear wave equations and dynamical systems along with the study on nonlinear wave equations by using the theory of dynamical systems are presented. In the end of the chapter, some preliminary knowledge of dynamical systems and the basic mathematic theory and main results of the three-step method are introduced.In Chapter 2, by improving a sub-equation method proposed by professor Engui Fan for finding exact solutions to nonlinear wave equations, the explicit exact solutions of the Sawada-Kotera equation are discussed. The essence of the sub-equation method is to convert the problem for finding travelling wave solutions of nonlinear wave equations to the problem for solving a sub-equation by making a wonderful polynomial transformation between the more complicated nonlinear wave equation and the simpler sub-equation. Thus how to find more exact solutions of the sub-equation becomes a key step in this method. Here by making full advantage of bifurcation theory of dynamical systems to study the sub-equation with general forms, an improved sub-equation method is presented and applied to the Sawada-Kotera equation. As a result, a series of travelling wave solutions to the Sawada-Kotera equation are obtained in a systematic way, which include multi-hump solitary wave solutions, multi-hump periodic wave solutions and so on. Moreover all the parameters in the obtained solutions are independent of the systematic parameters in the Sawada-Kotera equation, and therefore these solutions exhibit rich dynamical behavior by taking different values of these parameters. By using the improved method, not only all the solutions of the sub-equation with general forms can be gained (in the thesis, to save space, all solitary wave solutions and kink wave solutions, some rational function solutions along with some periodic travelling wave solutions are just given), but also the dynamical property and the parametric condition of each solution obtained can be known, which shows the advantage and effectiveness of the improved sub-equation method for researching exact solutions of nonlinear wave equations.In Chapter 3, the travelling wave solutions of a family of regularized long-wave equations, i.e., R(m,n) equations, are discussed by using the three-step method. After making a transformation of time scale, the singular travelling wave system of R(m,n) equations is reduced to a regular dynamical system. And the qualitative behavior of orbits of the regular system can be obtained by using the classical bifurcation theory of dynamical systems. Hence the qualitative information of the travelling wave system of R(m,n) equations are achieved from singular perturbation theory and the relation between the singular system and the regular system. How smooth travelling wave solutions lose their smoothness and become non-smooth travelling wave solutions is explained. And the fact is proved that the travelling wave solutions of R(m,n) equations corresponding to the singular homo-clinic orbits of the regular system are not their smooth solitary wave solutions but smooth periodic travelling wave solutions.In Chapter 4, a family of n+1-dimensional Klein-Gordon equations with nonlinear dissipative term and nonlinear dispersive term are studied. The effect on the system under the common actions of nonlinear dissipative intensity, nonlinear dispersive intensity and nonlinear intensity effect is discussed, which shows that the dynamical behavior of solutions depends greatly on the nonlinear intensity. It is emphasized that the existence of singular straight line is the original reason for the appearance of non-smooth periodic cusp wave solutions, solitary cusp wave solutions and breaking wave solutions. Various sufficient conditions to guarantee the existence of smooth and non-smooth travelling wave solutions are given. The singular travelling wave system and the regular travelling wave system have distinct time scales which cause their some corresponding orbits to have distinct dynamical properties. The orbits of the singular travelling wave system which correspond to the singular homoclinic orbits of the regular travelling wave system, for example, are either its periodic orbits or still homoclinic orbits. While the two distinct orbits respectively correspond to different solutions of Klein-Gordon equations with distinct dynamical behavior: the periodic orbits of singular system correspond to the periodic travelling wave solutions while the homoclinic orbits correspond to the solitary wave solutions. But how to judge the singular homoclinic orbits are periodic orbits or homoclinic orbits of singular system depends on the nonlinear dissipative intensity, nonlinear dispersive intensity and nonlinear intensity effect. These phenomena show sufficiently the essential effect on the nonlinear system under the three kinds of intensity. In this chapter, by taking advantage of singular perturbation theory, the wonderful phenomena that the corresponding orbits between the singular system and the regular system have different dynamical behavior are explained and strictly proved in mathematics. And the methods for judging the orbital dynamical property are given. These results obtained here enrich and develop the three-step method.In Chapter 5, the qualitative behavior of travelling wave solutions to two variants of 2+1 dimensional Boussinesq-type equations with positive and negative exponents respectively is studied. Since both their traveling wave systems have singularity, after applying qualitative theory of differential equations to the corresponding regular systems, the qualitative properties of all bounded orbits of the regular systems are obtained. Therefore the bifurcation parameter conditions which lead to smooth and non-smooth traveling wave solutions to the two variants of Boussinesq-type equations are analyzed and the various sufficient conditions to guarantee the existence of the bounded traveling wave solutions are obtained. Especially, for the Boussinesq-type equations with negative exponent, all the smooth orbits of the regular system correspond to the smooth orbits of the singular system. The singular homoclinic orbits and heteroclinic orbits of the regular system also are respectively the homoclinic orbits and heteroclinic orbits of the singular system, i.e., the smooth solitary wave solutions and smooth kink (or anti-kink) wave solutions of the Boussinesq-type equations with negative exponent. Hence the existence of uncountably infinite many smooth solitary wave solutions under certain parametric conditions is obtained. For the Boussinesq-type equations with negative exponent, the singularity does not cause the appearance of non-smooth traveling wave solutions, which shows that singular line does not always result in non-smooth solutions. That is to say, singular traveling wave systems do not always exist non-smooth traveling wave solutions.In Chapter 6, the summary of this dissertation and the prospect of future study are given.