A stabilizing effect of advection on planar interfaces in singularly perturbed reaction-diffusion equations

TL;DR

This paper investigates how advection influences the stability of planar traveling fronts in singularly perturbed reaction-diffusion equations, revealing a stabilizing effect at high advection levels and identifying a critical scaling for instability onset.

Contribution

It demonstrates that large advection coefficients stabilize fronts and identifies a critical scaling law for the onset of instability in reaction-diffusion-advection systems.

Findings

High advection stabilizes fronts against long wavelength perturbations.

A critical scaling $ u o u_c o ext{const} imes \, ext{delta}^{-4/3}$ marks the instability threshold.

Application to dryland ecosystem models illustrates practical relevance.

Abstract

We consider planar traveling fronts between stable steady states in two-component singularly perturbed reaction-diffusion-advection equations, where a small quantity $\delta^2$ represents the ratio of diffusion coefficients. The fronts under consideration are large amplitude and contain a sharp interface, induced by traversing a fast heteroclinic orbit in a suitable slow fast framework. We explore the effect of advection on the spectral stability of the fronts to long wavelength perturbations in two spatial dimensions. We find that for suitably large advection coefficient $\nu$, the fronts are stable to such perturbations, while they can be unstable for smaller values of $\nu$. In this case, a critical asymptotic scaling $\nu\sim \delta^{-4/3}$ is obtained at which the onset of instability occurs. The results are applied to a family of traveling fronts in a dryland ecosystem model.