Study On Bounded Traveling Wave Solutions Of A Class Of Nonlinear Wave Equations

Nonlinear wave equation is one of the important branches in nonlinear science. As a hot topic of research in nonlinear science, study on the method for finding solutions of nonlinear wave equation has become more and more challenging. Study on exact solutions of nonlinear wave equation is helpful in clarifying the underlying algebraic structure of the soliton theory and plays an important role in reasonable explaining of the corresponding natural phenomenon and application. Because of the complicity of nonlinear wave equation, there has no systematic methodology of finding solutions now. Many effective methods which have been used so far, can not ensure the boundedness of the given solutions, we then only understand the solutions partially. In this paper, the first integral method and the bifurcation methods of planar dynamical systems are used to investigate the bounded traveling wave solutions for nonlinear wave equations, which include the Degasperis-Procesi equation, the Degasperis-Procesi equation with the dispersion term and the two-component Degasperis-Procesi equation. By making use of the method proposed, the parameter variation are analyzed for the existence of the different-type bounded traveling wave solutions. The basic content of this paper are given as following:In the first chapter, the background of nonlinear wave equation is reviewed. Then the study development and significance of our equations are introduced. Also, the research arrangement of the dissertation is given in the end of this chapter.In the second chapter, the related knowledge of solitary waves is outlined and several important methods for studying soliton solutions are introduced. Finally, the concerned concepts and theories used in this paper are introduced.In the third chapter, the exact traveling wave solutions of the Degasperis-Procesi equation and the Degasperis-Procesi equation with the dispersion term are investigated via the first integral method which is based on the ring theory of commutative algebra, and our results extend the previously known results.In the fourth chapter, it is well known that the parameters influence the types of the solution of the system greatly. Bifurcation analysis has been introduced to understand the effect of the parameters on the solutions of the system better. By using the bifurcation method of planar dynamical system, we change the DP equation with the dispersion term into the traveling wave system and draw the bifurcation of phase portraits. Then periodic wave solutions and solitary wave solutions are constructed, and their convergence is showed when parameters vary. The peaked solitary wave solutions are obtained, and the reasons for the appearance of peaked points are discussed. In the end, graph of the solution is given under some parameter conditions.In the fifth chapter, the bifurcation method of planar dynamic systems is used to investigate bounded traveling wave solutions for the two component Degasperis-Procesi equation. By using the phase portrait bifurcation of traveling wave system, solitary wave solutions, periodic wave solutions and breaking kink (anti-kink) wave solutions are discussed under different parameter conditions. And loop soliton solution is also discussed. By using numerical simulation and mathematical software, the graph of some solutions is made under specific parameter conditions.Finally, the summary of this dissertation and the prospect of study on the nonlinear evolution equations are given.