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Snakes and ladders: localized states in the Swift-Hohenberg equation. (English) Zbl 1236.35144
Summary: 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.

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
35K55 Nonlinear parabolic equations
37L10 Normal forms, center manifold theory, bifurcation theory for infinite-dimensional dissipative dynamical systems
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