Some Problems On Bifurcations Of Nonlinear Evolution Equations

Nonlinear dynamics is one of the important branches in nonlinear science, whereas as a leading subject and hot interest in nonlinear science, study on the method for finding solutions of nonlinear partial differential equations has become more and more challenging. Because of the complicity in nonlinear evolution equations, there has no systemic and uniform method for all NLEEs. Many effective methods, such as many kinds of methods for exact solutions, numerical simulations and experimental ways, have been found, but we can only understand the solutions partly and not make sure of overall situations by all these methods. In this paper, we consider all possible bounded traveling wave solutions of some important nonlinear evolution equations. Bifurcation theories are used to analyse the parameter conditions for the existence of solitary wave solutions. The basic content of this paper are given as following:Chapter one is devoted to reviewing the history and development of the NLEEs.Bifurcations of KdV equation, Camassa-Holm equation and coupled Bousinesq equations are researched respectively in chapter two. Transition boundaries and phase portraits in different regions are given, based on which we can obtain all possible traveling wave solutions as well as the parameter conditions.In chapter three, we think about non-analytical (non-smooth) wave solutions. To a general KdV equation with singular curves on phase plane, we explain the reason why these non-smooth traveling wave solutions arise.Based on the bifurcation analysis, all possible traveling wave solutions have been known qualitatively. Then we find out the exact bounded wave solutions for WBK equation in chapter four. According to the theory of dynamical system, we investigate the explicit exact traveling wave solutions of nonlinear wave equations by using the characters of the closed trajectory connecting equilibrium points and the relations between obits and traveling waves. When parameters are taken in the same region, there may exist different types of solutions. Bifurcation mechanism between these solutions are revealed.Then we focus on interactions between different waves. A new method is proposed in chapter five. By nonlinear superposition of different single-mode waves, new types of multiple-mode waves can be derived. Several cases for the two-mode waves are obtained upon using the computer language Maple.Steady wave solutions with constant velocity, amplitude and width have fully been understood. Now we investigate the dynamical behavior of the cubic-quintic complex Ginzburg-Landau equation in this paper. Based on the assumption of a special trial function, a three-dimensional vector field has been derived from the infinite-dimensional dissipative system. Numerical simulations are used to reveal the complexity of the vector field.By making use of the approaches proposed by us, a variety of exact solutions to many significant nonlinear evolution equations are easily presented. Finally, the summary of this dissertation and the prospect of study on the nonlinear evolution equations are given.