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

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.