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

Montie Avery; Arnd Scheel

doi:10.1007/s10884-021-10093-3

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

Montie Avery, Arnd Scheel

School of Mathematics

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

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.

Original language	English (US)
Journal	Journal of Dynamics and Differential Equations
DOIs	https://doi.org/10.1007/s10884-021-10093-3
State	Accepted/In press - 2021

Bibliographical note

Funding Information:
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.

Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

Keywords

Diffusive stability
Ginzburg–Landau equation
Pulled fronts
Traveling waves

Publisher link

10.1007/s10884-021-10093-3

Cite this

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title = "Sharp Decay Rates for Localized Perturbations to the Critical Front in the Ginzburg–Landau Equation",

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.",

keywords = "Diffusive stability, Ginzburg–Landau equation, Pulled fronts, Traveling waves",

author = "Montie Avery and Arnd Scheel",

note = "Funding Information: 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. Publisher Copyright: {\textcopyright} 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.",

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journal = "Journal of Dynamics and Differential Equations",

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T1 - Sharp Decay Rates for Localized Perturbations to the Critical Front in the Ginzburg–Landau Equation

AU - Avery, Montie

AU - Scheel, Arnd

N1 - Funding Information: 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. Publisher Copyright: © 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2021

Y1 - 2021

N2 - 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.

AB - 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.

KW - Diffusive stability

KW - Ginzburg–Landau equation

KW - Pulled fronts

KW - Traveling waves

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

Abstract

Bibliographical note

Keywords

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