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

Ryan Goh, Arnd Scheel

Research output: Contribution to journalArticlepeer-review

13 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)104-128
Number of pages25
JournalJournal of the London Mathematical Society
Issue number1
StatePublished - Aug 2018

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© 2018 London Mathematical Society


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