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

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.