2017/08/12 by Ryan Goh, Arnd Scheel · 1 citation
Physics and Astronomy · Mathematics · #nlin.PS #math.AP #math.DS #msc:35B36 #msc:35B25 #msc:37L10 #msc:70K44 #msc:74N05 #msc:92C15 #msc:47A52
paper · pdf · doi:10.1112/jlms.12122
arxiv created 2017/08/12 · arxiv updated 2018/04/04
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.