Abstract
A new class of pattern forming systems is identified and investigated: anisotropic systems that are spatially inhomogeneous along the direction perpendicular to the preferred one. By studying the generic amplitude equation of this new class and a model equation, we show that branched stripe patterns emerge, which for a given parameter set are stable within a band of different wave numbers and different numbers of branching points (defects). Moreover, the branched patterns and unbranched ones (defect-free stripes) coexist over a finite parameter range. We propose two systems where this generic scenario can be found experimentally, surface wrinkling on elastic substrates and electroconvection in nematic liquid crystals, and relate them to the findings from the amplitude equation.
Highlights
Pattern formation is one of the most fascinating and intriguing phenomena in nature [1, 2]
Each picture is variation of the pattern’s natural wave number q0. This can be accomplished by varying the elasticity in the wrinkle forming system or the height of the electroconvection cell, respectively. In this class of anisotropic systems we find straight stripes, see figure 1(c), that are stable for wave numbers in a finite range around q 0, similar as in homogeneous anisotropic systems
In the following we present the universal amplitude equation of this new symmetry class of patterns and analyze its solutions
Summary
Pattern formation is one of the most fascinating and intriguing phenomena in nature [1, 2]. This can be accomplished by varying the elasticity in the wrinkle forming system or the height of the electroconvection cell, respectively In this class of anisotropic systems we find straight stripes, see figure 1(c), that are stable for wave numbers in a finite range around q 0, similar as in homogeneous anisotropic systems. The branched patterns coexist with the straight stripes in a wide parameter range This behavior is a non-trivial generalization of the wave number bands (Eckhaus bands) for homogeneous systems [37,38,39,40,41,42,43]—a well established concept and experimentally verified e.g. in EHC [39] and axisymmetric Taylor vortex flow [41, 42]—to inhomogeneous systems and multiple patterns. In the following we present the universal amplitude equation of this new symmetry class of patterns and analyze its solutions
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