Traveling wave solutions to nonlinear wave equations: dynamical system approach. Lecture 1

The dynamical system approach has been well applied to study the traveling wave solutions of various nonlinear wave equations in recent two decades. In this talk, the main ideal and the key steps of this approach to investigate the traveling wave solutions of nonlinear wave equations will be introduced firstly. In addition to the exact solutions, the bifurcations of the traveling wave solutions and various smooth or non-smooth can be derived via this approach. The advantage and the challenging of this method is discussed as well. How this approach is extended to examine the traveling waves solutions, even the multi-wave solutions of higher-order nonlinear wave equations will be presented. Some recent development and works on the persistence of traveling wave solutions and the new solitary wave solutions to some classical nonlinear wave equations under singular perturbation via combining the singular perturbation method and bifurcation analysis are presented.