Abstract
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 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.
Highlights
Pattern formation most commonly occurs with a single wavelength, as in for example zebra stripes, Rayleigh–Benard convection and the Taylor–Couette flow (Hoyle, 2006)
It has been recognised that pattern formation with two length scales can lead to a wide variety of complex and interesting patterns, such as superlattice patterns, quasipatterns and spatiotemporal chaos
Having two length scales can arise in different ways: in the Faraday wave problem with multi-frequency forcing, for example, patterns with the two length scales arise in response to different components of the forcing (Edwards & Fauve, 1994; Topaz & Silber, 2002; Rucklidge & Silber, 2009; Skeldon & Rucklidge, 2015)
Summary
Pattern formation most commonly occurs with a single wavelength, as in for example zebra stripes, Rayleigh–Benard convection and the Taylor–Couette flow (Hoyle, 2006). For the remainder of this paper, we will concentrate mostly on the parameter values where the marginal stability curves have two minima and there is bistability between patterns of different wavelength, leading to the possibility of these patterns coexisting in separate parts of the domain These equations are related to the two length scale models of Lifshitz & Petrich (1997) and Rucklidge et al (2012), our model cannot be derived from these by setting the two length scales to be equal. 3. Weakly nonlinear analysis we compute weakly nonlinear solutions for the model (2.5) by deriving a generalised version of the Ginzburg–Landau equation, and use it to establish where one-dimensional periodic patterns are stable to long-wave perturbations.
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