Sharp decay rates for localized perturbations to the critical front in the Ginzburg-Landau equation

TL;DR

This paper refines the understanding of the nonlinear stability of the critical invasion front in the Ginzburg-Landau equation, demonstrating faster decay rates for localized perturbations and detailed spectral analysis techniques.

Contribution

It provides sharper decay rate estimates for localized perturbations and introduces a resolvent analysis method near the essential spectrum for the Ginzburg-Landau equation.

Findings

Localized perturbations decay at rate t^{-3/2}

Phase perturbations decay diffusively

Refinement of previous stability results

Abstract

We revisit the nonlinear stability of the critical invasion front in the Ginzburg-Landau equation. Our main result shows that the amplitude of localized perturbations decays with rate $t^{-3/2}$, while the phase decays diffusively. We thereby refine earlier work of Bricmont and Kupiainen as well as Eckmann and Wayne, who separately established nonlinear stability but with slower decay rates. On a technical level, we rely on sharp linear estimates obtained through analysis of the resolvent near the essential spectrum via a far-field/core decomposition which is well suited to accurately describing the dynamics of separate neutrally stable modes arising from far-field behavior on the left and right.