Analytical results for front pinning between an hexagonal pattern and a uniform state in pattern-formation systems

Abstract

In pattern-forming systems, localized patterns are states of intermediate complexity between fully extended ordered patterns and completely irregular patterns. They are formed by stationary fronts enclosing an ordered pattern inside an homogeneous background. In two dimensions, the ordered pattern is most often hexagonal and the conditions for fronts to stabilize are still unknown. In this letter, we show how the locking of these fronts depends on their orientation relative to the pattern. The theory rests on general asymptotic arguments valid when the spatial scale of the front is slow compared to that of the hexagonal pattern. Our analytical results are confirmed by numerical simulations with the Swift-Hohenberg equation, relevant to hydrodynamical and buckling instabilities, and a nonlinear optical cavity model.

Cited by in corpus (11)

• Localized states in periodically forced systems
• Snakes and ghosts in a parity-time-symmetric chain of dimers
• Snaking branches of planar BCC fronts in the 3D Brusselator
• Localized patterns in planar bistable weakly coupled lattice systems
• Snakes in square, honeycomb, and triangular lattices
• Two-dimensional localized structures in harmonically forced oscillatory systems
• Dissecting the snake: the transition from localized patterns to isolated spikes in pattern formation systems
• Pattern formation with pde2path -- a tutorial
• On the speed of wave packets and the nonlinear Schrödinger equation
• Predicting the emergence of localised dihedral patterns in models for dryland vegetation
• Approximate localised dihedral patterns near a Turing instability