Invasion into remnant instability: a case study of front dynamics
Abstract
We study the invasion of an unstable state by a propagating front in a peculiar but generic situation where the invasion process exhibits a remnant instability. Here, remnant instability refers to the fact that the spatially constant invaded state is linearly unstable in any exponentially weighted space in a frame moving with the linear invasion speed. Our main result is the nonlinear asymptotic stability of the selected invasion front for a prototypical model coupling spatio-temporal oscillations and monotone dynamics. We establish stability through a decomposition of the perturbation into two pieces: one that is bounded in the weighted space and a second that is unbounded in the weighted space but which converges uniformly to zero in the unweighted space at an exponential rate. Interestingly, long-time numerical simulations reveal an apparent instability in some cases. We exhibit how this instability is caused by round-off errors that introduce linear resonant coupling of otherwise non-resonant linear modes, and we determine the accelerated invasion speed.
Keywords
Cite
@article{arxiv.2009.02106,
title = {Invasion into remnant instability: a case study of front dynamics},
author = {Gregory Faye and Matt Holzer and Arnd Scheel and Lars Siemer},
journal= {arXiv preprint arXiv:2009.02106},
year = {2020}
}