Snakes and ladders: Localized states in the Swift–Hohenberg equation

The Swift–Hohenberg equation with cubic and quintic nonlinearities exhibits multiple stable and unstable spatially localized states of arbitrary length in the vicinity of the Maxwell point between spatially homogeneous and periodic states. The even and odd states are organized in a characteristic snaking structure and are connected by branches of mixed parity states forming a ladder-like structure. Numerical computations are used to illustrate the changes in the localized solutions as they grow in spatial extent and to determine the stability and wavelength of the resulting states.