The main purpose of this lecture is to accomplish the existence and stability of a fast traveling pulse solution as well as the existence and instability of a slow traveling pulse solution of a nonlinear singularly perturbed system of reaction diffusion equations with a Heaviside step function. These equations are also called the discontinuous Fitzhugh-Nagumo equations.

 

The author will couple together implicit function theorem and many techniques in dynamical systems to establish the existence of the traveling pulse solutions and he will construct Evans functions and uses a global strong maximum principle for complex analytic functions to accomplish the stability of the fast traveling pulse solution and the instability of the slow traveling pulse solution.