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Sharp Decay Rates for Localized Perturbations to the Critical Front in the Ginzburg–Landau Equation

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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.

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Acknowledgements

This material is based upon work supported by the National Science Foundation through the Graduate Research Fellowship Program under Grant No. 00074041, as well as through NSF-DMS-1907391. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

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  1. School of Mathematics, University of Minnesota, 206 Church St. S.E., Minneapolis, MN, 55455, USA

    Montie Avery & Arnd Scheel

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  1. Montie Avery
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  2. Arnd Scheel
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Correspondence to Arnd Scheel.

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Dedicated to the memory of Pavol Brunovský

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Avery, M., Scheel, A. Sharp Decay Rates for Localized Perturbations to the Critical Front in the Ginzburg–Landau Equation. J Dyn Diff Equat 36 (Suppl 1), 287–322 (2024). https://doi.org/10.1007/s10884-021-10093-3

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  • DOI: https://doi.org/10.1007/s10884-021-10093-3

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