Large-amplitude modulation of periodic traveling waves

Guy Métivier; Kevin Zumbrun

doi:10.3934/dcdss.2022070

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2022, Volume 15, Issue 9: 2609-2632. Doi: 10.3934/dcdss.2022070

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Large-amplitude modulation of periodic traveling waves

Guy Métivier^1, and
Kevin Zumbrun^2, ,

1.
Université Bordeaux 1, Talence, France
2.
Indiana University, Bloomington, IN 47405, USA

^* Corresponding author: Kevin Zumbrun
^* Corresponding author: Kevin Zumbrun

Received: November 2021

Revised: February 2022

Early access: March 2022

Published: September 2022

Research of K.Z. was partially supported under NSF grant no. DMS-0300487

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Abstract

We introduce a new approach to the study of modulation of high-frequency periodic wave patterns, based on pseudodifferential analysis, multi-scale expansion, and Kreiss symmetrizer estimates like those in hyperbolic and hyperbolic-parabolic boundary-value theory. Key ingredients are local Floquet transformation as a preconditioner removing large derivatives in the normal direction of background rapidly oscillating fronts and the use of the periodic Evans function of Gardner to connect spectral information on component periodic waves to block structure of the resulting approximately constant-coefficient resolvent ODEs. Our main result is bounded-time existence and validity to all orders of large-amplitude smooth modulations of planar periodic solutions of multi-D reaction diffusion systems in the high-frequency/small wavelength limit.

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Mathematics Subject Classification: Primary: 35K57; Secondary: 35A35, 35Q53.

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