Title: 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.