Orbital stability of monostable waves for reaction-diffusion systems

TL;DR

This paper investigates the stability of monostable traveling waves in reaction-diffusion systems, demonstrating convergence to shifted profiles under certain initial conditions using explicit resolvent kernel estimates.

Contribution

It introduces a novel stability analysis method that handles weakly localized perturbations and phase shifts without relying on Evans function zeros.

Findings

Proves convergence to shifted wave profiles for initial data close to the wave

Develops explicit resolvent kernel estimates for stability analysis

Addresses phase shift construction without Evans function zero assumptions

Abstract

We study stability of monostable waves for reaction-diffusion systems. When the solution is initially close to a fast wave profile in optimal topology, we prove convergence to a shifted profile. The proof relies on explicit resolvent kernels estimates, allowing to handle weakly localized perturbations. It allows phase shift construction even when the translational eigenvalue is not associated to a zero of the Evans function. We further discuss distinction between Evans and Fourier eigenmodes when the marginal group velocity are directed towards the wave interface.