Localized dynamic patterns in the complex cubic–quintic Swift–Hohenberg equation in one and two dimensions

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We use bifurcation analysis and numerical methods to study dynamic patterns in a complex Swift-Hohenberg equation focusing on structures present on a 1D interval with periodic boundary conditions and on a 2D disk with Neumann boundary conditions. In 1D, the equation features Hopf bifurcations from the trivial state u ≡ 0 at finite wave number k ≠ 0 (also known as wave bifurcations), leading to a bifurcation problem with O(2)×S1 symmetry, i.e., symmetry under translations and reflections in space, and phase rotations. 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 some of the solutions thus obtained exhibit homoclinic snaking associated with their gradual growth in spatial extent. We exploit gauge symmetry to compute these dynamic patterns as relative equilibria, which allows us to locate efficiently tertiary bifurcations to two-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 2D 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.