Pattern-forming fronts in a Swift-Hohenberg equation with directional quenching - parallel and oblique stripes

TL;DR

This paper investigates how domain growth influences the orientation of stripe patterns in a Swift-Hohenberg equation, constructing solutions for parallel and oblique stripes using advanced mathematical techniques.

Contribution

It introduces novel methods to construct stripe-front solutions at various angles, addressing challenges like continuous spectrum and singular perturbations.

Findings

Constructed parallel stripe solutions using infinite-dimensional spatial dynamics.

Developed a perturbative approach for oblique stripe formation.

Resolved spectral and perturbation challenges with advanced analytical tools.

Abstract

We study the effect of domain growth on the orientation of striped phases in a Swift-Hohenberg equation. Domain growth is encoded in a step-like parameter dependence that allows stripe formation in a half plane, and suppresses patterns in the complement, while the boundary of the pattern-forming region is propagating with fixed normal velocity. We construct front solutions that leave behind stripes in the pattern-forming region that are parallel to or at a small oblique angle to the boundary. Technically, the construction of stripe formation parallel to the boundary relies on ill-posed, infinite-dimensional spatial dynamics. Stripes forming at a small oblique angle are constructed using a functional-analytic, perturbative approach. Here, the main difficulties are the presence of continuous spectrum and the fact that small oblique angles appear as a singular perturbation in a traveling-wave problem. We resolve the former difficulty using a farfield-core decomposition and Fredholm theory in weighted spaces. The singular perturbation problem is resolved using preconditioners and boot-strapping.