Front propagation close to the onset of instability
Abstract
We describe the resulting spatiotemporal dynamics when a homogeneous equilibrium loses stability in a spatially extended system. More precisely, we consider reaction-diffusion systems, assuming only that the reaction kinetics undergo a transcritical, saddle-node, or supercritical pitchfork bifurcation as a parameter passes through zero. We construct traveling front solutions which describe the invasion of the now-unstable state by a nearby stable state. We show that these fronts are marginally spectrally stable near the bifurcation point, which, together with recent advances in the theory of front propagation into unstable states, establishes that these fronts govern the dynamics of localized perturbation to the unstable state. Our proofs are based on functional analytic tools to study the existence and eigenvalue problems for fronts, which become singularly perturbed after a natural rescaling.
Keywords
Cite
@article{arxiv.2310.13602,
title = {Front propagation close to the onset of instability},
author = {Montie Avery},
journal= {arXiv preprint arXiv:2310.13602},
year = {2023}
}
Comments
This material originally appeared as part of arXiv:2211.11829 by the same author. At the suggestion of the anonymous referee of arXiv:2211.11829, we have separated this material into its own manuscript