Nonlinear stability of source defects in oscillatory media
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Citation (published version)Margaret Beck, Toan Nguyen, Bjorn Sandstede, Kevin Zumbrun. "Nonlinear stability of source defects in oscillatory media." Preprint, on arXiv https://arxiv.org/abs/1802.07676
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.