Localized patterns in a generalized Swift–Hohenberg equation with a quartic marginal stability curve

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 organized 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 localized in a background of patterns of another wavelength. In the small-amplitude limit, the amplitude equation for the model is a generalized Ginzburg–Landau equation with fourth-order spatial derivatives, which can take the form of a complex Swift–Hohenberg equation with real coefficients. Localized solutions in this amplitude equation help interpret the localized 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.