Existence of Traveling Wave Solutions to Reaction-Diffusion-ode Systems with Hysteresis

This paper establishes the existence of traveling wave solutions to a reaction-diffusion equation coupled with a singularly perturbed first order ordinary differential equation with a small parameter ϵ>0. The system is a toy model for biological pattern formation. Traveling wave solutions correspond to heteroclinic orbits of a fast-slow system. Under some conditions, the reduced problem (with ϵ=0) has a heteroclinic orbit with jump discontinuity, while the layer problem (i.e., the fast subsystem obtained as another limit of ϵ→0) has an orbit filling the gap. We thus construct a singular orbit by piecing together these two orbits. The traveling wave solution is obtained in the neighborhood of the singular orbit. However, unlike the classical FitzHugh-Nagumo equations, the singular orbit contains a fold point where the normal hyperbolicity breaks down and the standard Fenichel theory is not applicable. To circumvent this difficulty we employ the directional blowup method for geometric desingularization around the fold point.