Defect-like structures and localized patterns in SH357

Abstract

We study numerically the cubic-quintic-septic Swift-Hohenberg (SH357) equation on bounded one-dimensional domains. Under appropriate conditions stripes with wave number $k\approx 1$ 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 defect-like 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.

12 pages, 9 figures

References in corpus (6)

• Chimera States in Continuous Media: Existence and Distinctness
• Routes to synchrony between asymmetrically interacting oscillator ensembles
• Snaking branches of planar BCC fronts in the 3D Brusselator
• Localized radial roll patterns in higher space dimensions
• Tristability between stripes, up-, and down-hexagons and snaking bifurcation branches of spatial connections between up- and down-hexagons
• The origin of localized snakes-and-ladders solutions of plane Couette flow

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• Localised patterns in a generalised Swift--Hohenberg equation with a quartic marginal stability curve
• Towards Convectons in the Supercritical Regime: Homoclinic Snaking in Natural Doubly Diffusive Convection
• The origin of localized snakes-and-ladders solutions of plane Couette flow
• Snakes on Lieb lattice
• Pattern formation with pde2path -- a tutorial
• Modified snaking in plane Couette flow with wall-normal suction
• Snaking without subcriticality: grain boundaries as non-topological defects