The Bifurcations Of Nonlinear Waves For The Gardner Equation

Nonlinear wave equations are a class of important mathematical models describing a number of natural phenomenon, are also one of the most important study object in Mathematics, Physics, especially the soliton theory. By studying travelling wave solu-tions, numerical solutions and stability of solutions for nonlinear wave equations, people can gradually make clear the motion laws of the movement of the nonlinear systems under the nonlinear interactions, explain natural phenomena, descript essential characteristic-s of the system profoundly. This greatly promotes developments of related disciplines such as Physics, Mechanics, Applied Mathematics and Engineering Technologies. In the present thesis, the typical nonlinear wave equation, i.e. the Gardner equation is solved based on the qualitative theory of differential equations and the bifurcation method of dynamical systems. The exact traveling wave solutions and their interesting bifurcation phenomena are studied, and a series of new results are obtained. The concrete research work of this dissertation as follows:In Chapter 1, firstly, we give a simple summary of the historical background and the solving methods of the nonlinear wave equations. Secondly, we introduce the solving methods of the nonlinear wave equations and give a simple summary of the historical background and areas of application of the qualitative theory of differential equations and the bifurcation method of dynamical systems. We demonstrate the specific operation of the method by using the mKdV equation. Finally, we present the major works of this dissertation.In Chapter 2, we study the exact traveling wave solutions for the Gardner equation via the dynamical systems theory, firstly, we show the bifurcation phases portraits for the corresponding plane system of this equation. Secondly, under different parameter conditions, the new exact traveling wave solutions of the Gardner equation are obtained, including solitary wave solutions, periodic wave solutions, blow-up wave solutions, peri-odic blow-up wave solutions and kink wave solutions. Finally, we verify the correctness of the solutions via Software Mathematica.In Chapter 3, we make the further analysis to the exact traveling wave solutions of the Gardner equation and find out some interesting bifurcation phenomena, that is, the solitary waves can be bifurcated from the periodic waves; the kink waves can be bifurcated from the solitary waves and the blow-up waves; the blow-up waves can be bifurcated from the solitary waves; the periodic blow-up waves can be bifurcated from the periodic waves, and the concrete variation of those traveling waves are shown in the figures.Chapter 4 is the summary and prospect of this dissertation.