/ October 5, 2022

Date: Wed. October 5, 2022
Event: FAU DCN-AvH Seminar
Organized by: FAU DCN-AvH, Chair for Dynamics, Control and Numerics – Alexander von Humboldt Professorship at FAU Friedrich-Alexander-Universität Erlangen-Nürnberg (Germany)
Title: FAU DCN-AvH Seminar “Large-amplitude modulation of periodic traveling waves”

Speaker: Prof. Dr. Kevin Zumbrun
Affiliation: Department of Mathematics, Indiana University (USA)

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. This result is new for either large amplitudes or multi-d. Applications to pattern formation are many, and well known (Howard-Kopell, Doelman-Sandstede-Scheel-Schneider,…). The method suggests possibilities also to treat more complicated phenomena in shallow-water and thin-film flow. The talk is based on a joint work with G. Metivier.

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