Localised patterns in a generalised Swift--Hohenberg equation with a quartic marginal stability curve

TL;DR

This paper investigates localized pattern formation near a codimension-three point with a quartic marginal stability curve using a generalized Swift--Hohenberg model, revealing snaking branches of localized patterns of different wavelengths.

Contribution

It introduces a generalized Swift--Hohenberg model with a quartic minimum in the stability curve, extending understanding of localized patterns and snaking behavior in pattern-forming systems.

Findings

Identification of snaking branches of localized patterns

Development of a generalized Ginzburg--Landau amplitude equation

Interpretation of localized solutions in the amplitude equation

Abstract

In some pattern-forming systems, for some parameter values, patterns form with two wavelengths, while for other parameter values, there is only one wavelength. The transition between these can be organised by a codimension-three point at which the marginal stability curve has a quartic minimum. We develop a model equation to explore this situation, based on the Swift--Hohenberg equation; the model contains, amongst other things, snaking branches of patterns of one wavelength localised in a background of patterns of another wavelength. In the small-amplitude limit, the amplitude equation for the model is a generalised Ginzburg--Landau equation with fourth-order spatial derivatives, which can take the form of a complex Swift--Hohenberg equation with real coefficients. Localised solutions in this amplitude equation help interpret the localised patterns in the model. This work extends recent efforts to investigate snaking behaviour in pattern-forming systems where two different stable non-trivial patterns exist at the same parameter values.