Diffusive stability against nonlocalized perturbations of planar wave trains in reaction-diffusion systems

TL;DR

This paper proves nonlinear diffusive stability of planar wave trains in reaction-diffusion systems against nonlocalized perturbations, using novel pointwise estimates and coordinate tracking methods.

Contribution

It is the first to analyze stability under spatially nonlocalized perturbations not arising from phase modulation.

Findings

Established nonlinear stability against line-bounded, exponentially decaying perturbations.

Achieved better decay rates for fully localized perturbations.

Developed a new analytical approach tracking solutions in two coordinate systems.

Abstract

Planar wave trains are traveling wave solutions whose wave profiles are periodic in one spatial direction and constant in the transverse direction. In this paper, we investigate the stability of planar wave trains in reaction-diffusion systems. We establish nonlinear diffusive stability against perturbations that are bounded along a line in $\mathbb{R}^2$ and decay exponentially in the distance from this line. Our analysis is the first to treat spatially nonlocalized perturbations that do not originate from a phase modulation. We also consider perturbations that are fully localized and establish nonlinear stability with better decay rates, suggesting a trade-off between spatial localization of perturbations and temporal decay rate. Our stability analysis utilizes pointwise estimates to exploit the spatial structure of the perturbations. The nonlocalization of perturbations prevents the use of damping estimates in the nonlinear iteration scheme; instead, we track the perturbed solution in two different coordinate systems.