Title: Traveling Pulses and Undulating Fronts in Two Nonlinear Systems

Abstract: In this talk I’ll discuss recent work on the asymptotic analysis of dynamic phenomena in two singularly perturbed nonlinear systems of (integro-) differential equations. First, we’ll consider an eco-evolutionary model of communicable disease in which infected and recovered populations are distributed over a one-dimensional antigenic space. A careful WKB-type analysis leads to an analytical tractable reduction with which endemic solutions taking the form of traveling pulses can be fully characterized. I’ll describe some of the key insights we gain from such an analysis as well as my main objectives for future research. The second part of this talk focuses on a minimal model of mass-conserved non-reciprocal dynamics recently introduced by Brauns and Marchetti. Following a careful asymptotic analysis in the sharp interface limit we obtain a modification of the classical Mullins-Sekerka system for interface evolution. With planar fronts as a test case, we illustrate how non-reciprocity can lead to rich nonlinear dynamics such as undulating traveling fronts. I’ll conclude by highlighting some of the numerical challenges involved in solving the sharp-interface equations, as well as some of the phenomena that the sharp interface limit may help us address.