Pattern formation in a Swift-Hohenberg equation with spatially periodic coefficients

TL;DR

This paper analyzes pattern formation in the Swift-Hohenberg equation with large periodic coefficients, revealing how resonant and non-resonant regimes influence bifurcations and pattern structures, extending previous small-coefficient results.

Contribution

It extends bifurcation analysis of the Swift-Hohenberg equation to large coefficients, directly working in Bloch space and distinguishing resonant effects, applicable to complex heterogeneous systems.

Findings

Identification of Turing bifurcation with Bloch wave patterns

Analysis of resonant vs. non-resonant regimes affecting pattern formation

Framework applicable to reaction-diffusion systems with heterogeneity

Abstract

We study the Swift-Hohenberg equation - a paradigm model for pattern formation - with "large" spatially periodic coefficients and find a Turing bifurcation that generates patterns whose leading order form is a Bloch wave modulated by solutions of a Ginzburg-Landau type equation. Since the interplay between forcing wavenumber and intrinsic wavenumber crucially shapes the spectrum and emerging patterns, we distinguish between resonant and non-resonant regimes. Extending earlier work that assumed asymptotically small coefficients, we tackle the more involved onset analysis produced by O(1) forcing and work directly in Bloch space, where the richer structure of the bifurcating solutions becomes apparent. This abstract framework is readily transferable to more complex systems, such as reaction-diffusion equations arising as dryland vegetation models, where topography induces spatial heterogeneity.