Abstract
Nonlinear stripe patterns in two spatial dimensions break the rotational symmetry and generically show a preferred orientation near domain boundaries, as described by the famous Newell–Whitehead–Segel (NWS) equation. We first demonstrate that, as a consequence, stripes favour rectangular over quadratic domains. We then investigate the effects of patterns ‘living’ in deformable domains by introducing a model coupling a generalized Swift–Hohenberg model to a generic phase field model describing the domain boundaries. If either the control parameter inside the domain (and therefore the pattern amplitude) or the coupling strength (‘anchoring energy’ at the boundary) are increased, the stripe pattern self-organizes the domain on which it ‘lives’ into anisotropic shapes. For smooth phase field variations at the domain boundaries, we simultaneously find a selection of the domain shape and the wave number of the stripe pattern. This selection shows further interesting dynamical behavior for rather steep variations of the phase field across the domain boundaries. The here-discovered feedback between the anisotropy of a pattern and its orientation at boundaries is relevant e.g. for shaken drops or biological pattern formation during development.
Highlights
Fascinating pattern formation phenomena are ubiquitous in nature [1–4]
For smooth phase field variations at the domain boundaries, we simultaneously find a selection of the domain shape and the wave number of the stripe pattern
In the often occurring case where the pattern forming field has to vanish outside the domain, this results in an orientational preference for the stripes with respect to domain boundaries
Summary
Fascinating pattern formation phenomena are ubiquitous in nature [1–4]. Examples include spatially periodic patterns in convection systems from small to geological scales [4–7], surface waves [8] or patterns in biological systems [9–11] down to the scales of single cells, including for instance cell polarization that resembles phase separation [12–14]. Examples are pattern forming light-sensitive chemical reactions [24], in vitro protein reactions on membranes [25] or again fluid systems [26–28] In these cases, no specific boundary conditions act on the concentration or flow fields that will become patterned but rather the control parameter drops from super- to subcritical values outside of a certain domain. No specific boundary conditions act on the concentration or flow fields that will become patterned but rather the control parameter drops from super- to subcritical values outside of a certain domain Such spatial restrictions have been so far mostly studied in quasi one-dimensional systems. Experiments [15, 16] and simulations [32] show that with increasing amplitude of the Faraday waves, the liquid drops of originally circular cross-section (their three-dimensional shape being a spherical cap) deform to elliptical and even worm-like shapes Is this a general effect for patterns in deformable domains? Understanding the generic properties of the ‘self-shaping’ of ‘active domains’ may give insights for the development of biomimetic approaches along the lines of, e.g. [25]
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