CSDS seminar – Dynamic patterns in the complex Swift-Hohenberg equation over intervals and disks – Hannes Uecker

We use bifurcation analysis and numerical methods to study dynamic patterns in the complex Swift-Hohenberg (CSH) equation over bounded intervals with periodic boundary conditions, and over disks with Neumann boundary conditions. The equation features spatially inhomogeneous Hopf bifurcations (aka wave bifurcations) from the trivial branch, leading to a bifurcation problem with O(2) x S^1 symmetry, i.e., symmetry under translations and reflections in space, and phase rotations (gauge). The O(2) symmetry implies the simultaneous appearance of traveling waves and standing waves, and we consider the case where both bifurcations are subcritical and yield secondary bifurcations to spatially localized structures in the form of modulated traveling  waves and localized standing waves, respectively. We use numerical continuation to show that in suitable parameter regimes some of the branches thus obtained exhibit homoclinic snaking associated with their gradual growth in spatial extent. We exploit the gauge symmetry to compute these dynamic patterns as relative equilibria, which allows to efficiently find tertiary bifurcations to 2-frequency localized standing waves, and to localized drifting waves. The intricate behavior identified in 1D is shown to provide a road map for the organization of wall-attached states on disks in the form of rotating and standing waves at the wall, which may further localize in angle, yielding rotating spots or stationary breathing spots at the wall, and tertiary bifurcations from these. Some direct numerical simulations are used to identify further structures, including dynamics present in the disk bulk.