Georgia Institute of Technology College of Sciences

Nonlinear wave phenomena are of great importance in the physical world and have been for a long time a challenging topic of research for both pure and applied mathematicians. There are numerous nonlinear evolution equations for which we need to analyze the properties of the solutions for time evolution of the system. As the first step, we should understand the dynamics of their traveling wave solutions. There exists an enormous literature on the study of nonlinear wave equations, in which exact explicit solitary wave, kink wave, periodic wave solutions and their dynamical stabilities are discussed. To find exact traveling wave solutions for a given nonlinear wave system, a lot of methods have been developed. What is the dynamical behavior of these exact traveling wave solutions? How do the travelling wave solutions depend on the parameters of the system? What is the reason of the smoothness change of traveling wave solutions? How to understand the dynamics of the so-called compacton and peakon solutions? These are very interesting and important problems. The aim of this talk is to give a more systematic account for the bifurcation theory method of dynamical systems to find traveling wave solutions and understand their dynamics for two classes of singular nonlinear traveling systems.