The Nonlinear Wave Solutions And Bifurcation For Several Kinds Of Partial Differential Equations

Partial differential equations are important mathematical models for describing natural phenomena and are one of the forefront topics in the studies of soliton theory. This doctoral dissertation is devoted to study the nonlinear wave solutions and bifurcation for several kinks of partial di?erential equations, such as generalized Zakharov equations,BBM-like B(m,n)equations, generalized Kd V equation and so on. By using qualitative theory of di?erential equation, the method of bifurcations of dynamical systems, symbolic computation, numerical simulation and so on, on one hand, we obtain solitary wave solutions, kink wave solutions, periodic wave solutions, blow-up wave solutions, kink-like wave solutions, compacton-like wave solutions for these equations; On the other hand, we obtain some bifurcation phenomena for these traveling wave solutions. The main research works of this dissertation are as follows.In Chapter 1, the historical background, research developments, the main methods and achievements for solving nonlinear partial di?erential equations are summarized.The relation between partial di?erential equations and dynamical systems along with the study are presented. In the end of the chapter, some preliminary knowledge of dynamical systems and the main results for studying partial di?erential equations are introduced,which are proposed by professor Li Jibin and professor Liu Zhengrong.In Chapter 2, using bifurcation method of dynamical systems, we investigate the nonlinear waves for the generalized Zakharov equations. We obtain the following results:(i)Three types of explicit expressions of nonlinear waves are obtained, that is, the fractional expressions, the trigonometric expressions and the exp-function expressions.(ii) Under di?erent parameter conditions, these expressions represent symmetric and anti-symmetric solitary waves, kink and anti-kink waves, symmetric periodic and periodic-blow-up waves,1-blow-up and 2-blow-up waves. We point out that there are two sets of kink waves which are called tall-kink waves and low-kink waves respectively.(iii) Five kinds of interesting bifurcation phenomena are revealed. The ?rst kind is that the 1-blow-up waves can be bifurcated from the periodic-blow-up and 2-blow-up waves. The second kind is that the2-blow-up waves can be bifurcated from the periodic-blow-up waves. The third kind is that the symmetric solitary waves can be bifurcated from the symmetric periodic waves.The fourth kind is that the low-kink waves can be bifurcated from four types of nonlinear waves, the symmetric solitary waves, the 1-blow-up waves, the tall-kink waves and the anti-symmetric solitary waves. The ?fth kind is that the tall-kink waves can be bifurcated from the symmetric periodic waves. We also show that the exp-function expressions include some results given by pioneers.In Chapter 3, we investigate the traveling wave solutions and their bifurcations for the BBM-like B(m, n) equations by using bifurcation method and numerical simulation approach of dynamical systems. Firstly, for BBM-like B(3, 2) equation, we obtain some precise expressions of traveling wave solutions, which include periodic blow-up and periodic wave solution, peakon and periodic peakon wave solution, solitary wave and blow-up solution. Furthermore, we reveal the relationships among these solutions theoretically.Secondly, for BBM-like B(4, 2) equation, we construct two periodic wave solutions and two blow-up solutions. In order to con?rm the correctness of these solutions, we also check them by software Mathematica.In Chapter 4, by using the bifurcation method of dynamical systems and numerical simulation approach of di?erential equations, we investigate generalized Kd V equation.Two types of bounded traveling wave solutions are found, that is, the kink-like wave and compacton-like wave solutions. The planar graphs of these solutions are simulated by using software Mathematica, meanwhile, some interesting phenomena are revealed,that is, under some conditions the periodic wave can become the kink-like wave and compacton-like wave respectively, the solitary wave can become the kink-like wave. Exact implicit or parameter expressions of the kink-like wave and compacton-like wave solutions are given.