Large-amplitude modulation of periodic traveling waves

TL;DR

This paper develops a new analytical framework combining pseudodifferential analysis, multi-scale expansion, and Floquet theory to study large-amplitude modulations of high-frequency periodic waves in reaction-diffusion systems, establishing bounded-time existence results.

Contribution

It introduces a novel approach integrating advanced mathematical tools to analyze large-amplitude wave modulations, extending the understanding of stability and existence in multi-D reaction-diffusion systems.

Findings

Proves bounded-time existence of large-amplitude modulations.

Connects spectral properties to block-structured resolvent ODEs.

Establishes validity of high-frequency modulation analysis.

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 valitidy 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.