Pushed fronts in a Fisher-KPP-Burgers system using geometric desingularization

TL;DR

This paper investigates traveling fronts in a reaction-diffusion-advection system inspired by reactive flows, using geometric blow-up techniques to handle non-hyperbolic points and determine wave speeds.

Contribution

It introduces a novel application of geometric desingularization to analyze traveling fronts in a Fisher-KPP-Burgers system with non-hyperbolic end states.

Findings

Constructed approximate front profiles in the large parameter limit.

Derived leading order expansion for the wavespeed.

Established existence of traveling fronts using geometric blow-up.

Abstract

We study traveling fronts in a system of one dimensional reaction-diffusion-advection equations motivated by problems in reactive flows. In the limit as a parameter tends to infinity, we construct the approximate front profile and determine the leading order expansion for the selected wavespeed. Such fronts are often constructed as transverse intersections of stable and unstable manifolds of the traveling wave differential equation. However, a re-scaling of the dependent variable leads to a lack of hyperbolicity for one of the end states making the definition of one such manifold unclear. We use geometric blow-up techniques to recover hyperbolicity and following an analysis of the blown up vector field are able to show the existence of a traveling front with a leading order expansion of its speed.