Nonlinear stability of source defects in oscillatory media

TL;DR

This paper establishes the nonlinear stability of spectrally stable time-periodic source defects in reaction-diffusion systems, demonstrating that localized perturbations decay and the system converges to a translated pattern.

Contribution

It introduces a novel analysis combining spatial dynamics and phase-tracking to prove stability and detailed perturbation behavior of source defects.

Findings

Perturbations radiate outward as phase-shifted waves at a linear rate.

Perturbed solutions converge exponentially to a space-time translate of the original pattern.

Detailed pointwise estimates describe the decay and propagation of perturbations.

Abstract

In this paper, we prove the nonlinear stability under localized perturbations of spectrally stable time-periodic source defects of reaction-diffusion systems. Consisting of a core that emits periodic wave trains to each side, source defects are important as organizing centers of more complicated flows. Our analysis uses spatial dynamics combined with an instantaneous phase-tracking technique to obtain detailed pointwise estimates describing perturbations to lowest order as a phase-shift radiating outward at a linear rate plus a pair of localized approximately Gaussian excitations along the phase-shift boundaries; we show that in the wake of these outgoing waves the perturbed solution converges time-exponentially to a space-time translate of the original source pattern.