Traveling Pulse Solutions in a Three-Component FitzHugh--Nagumo Model

We use geometric singular perturbation techniques combined with an action functional approach to study traveling pulse solutions in a three-component FitzHugh--Nagumo model. First, we derive the profile of traveling 1-pulse solutions with undetermined width and propagating speed. Next, we compute the associated action functional for this profile from which we derive the conditions for existence and a saddle-node bifurcation as the zeros of the action functional and its derivatives. We obtain the same conditions by using a different analytical approach that exploits the singular limit of the problem. We also apply this methodology of the action functional to the problem for traveling 2-pulse solutions and derive the explicit conditions for existence and a saddle-node bifurcation. From these we deduce a necessary condition for the existence of traveling 2-pulse solutions. We end this article with a discussion related to Hopf bifurcations near the saddle-node bifurcation. (A corrected version is attached.)