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.
Bibliographical noteFunding Information:
Received 12 August 2017; published online 30 March 2018. 2010 Mathematics Subject Classification 35B25, 35B36, 37L10, 47A52, 70K44 (primary), 74N05, 92C15 (secondary). This research was partially supported by the National Science Foundation through the grants NSF-DMS-1603416 (R. Goh), and NSF-DMS-1612441, NSF-DMS-1311740 (A. Scheel), as well as a UMN Doctoral Dissertation Fellowship (R. Goh).