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

TL;DR

This paper develops an analytical theory for the stability of stationary fronts in pattern-forming systems, focusing on how their orientation relative to hexagonal patterns affects front pinning, validated by numerical simulations.

Contribution

It introduces a new analytical approach to understand front pinning in hexagonal pattern systems, linking front orientation to stability conditions.

Findings

Front pinning depends on the orientation of the front relative to the hexagonal pattern.

Analytical results are validated by numerical simulations of the Swift-Hohenberg equation.

The theory applies to systems like hydrodynamical and optical pattern formations.

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.