Defectlike structures and localized patterns in the cubic-quintic-septic Swift-Hohenberg equation.

Abstract

We study numerically the cubic-quintic-septic Swift-Hohenberg (SH357) equation on bounded one-dimensional domains. Under appropriate conditions stripes with wave number k≈1 bifurcate supercritically from the zero state and form S-shaped branches resulting in bistability between small and large amplitude stripes. Within this bistability range we find stationary heteroclinic connections or fronts between small and large amplitude stripes, and demonstrate that the associated spatially localized defectlike structures either snake or fall on isolas. In other parameter regimes we also find heteroclinic connections to spatially homogeneous states and a multitude of dynamically stable steady states consisting of patches of small and large amplitude stripes with different wave numbers or of spatially homogeneous patches. The SH357 equation is thus extremely rich in the types of patterns it exhibits. Some of the features of the bifurcation diagrams obtained by numerical continuation can be understood using a conserved quantity, the spatial Hamiltonian of the system.