Pattern-forming fronts in a Swift-Hohenberg equation with directional quenching - parallel and oblique stripes
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.
Keywords
Cite
@article{arxiv.1708.03815,
title = {Pattern-forming fronts in a Swift-Hohenberg equation with directional quenching - parallel and oblique stripes},
author = {Ryan Goh and Arnd Scheel},
journal= {arXiv preprint arXiv:1708.03815},
year = {2018}
}