Orbital stability of monostable waves for reaction-diffusion systems

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.