Coexistence of Stable Branched Patterns in Anisotropic Inhomogeneous Systems

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 wavenumbers 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.


Introduction
Pattern formation is one of the most fascinating and intriguing phenomena in nature [1,2]. It takes place in a wide variety of physical, chemical and biological systems and on disparate spatial and temporal scales, for example, convection phenomena in geoscience [2,3] or in liquid crystals [4][5][6], environmental patterns [7][8][9], or patterns in chemical reactions [10,11] and bacterial colonies [12]. In some circumstances pattern formation is undesired, for instance, the formation of spiral waves leading to cardiac arrhythmias in the heart muscle [13]. In other contexts pattern formation is even essential for the functioning of a system, e.g. in embryo development [14] or when designing surface wrinkling patterns to fabricate nanometer-scale structures [15,16]. The mechanisms leading to the same type of pattern in different systems are obviously very diverse. Nevertheless, patterns occuring in systems of the same symmetry share common qualitative properties that can be described by universal amplitude equations for the envelope of periodic patterns [2,5,6,17].
Generating, modifying or eliminating patterns hence either requires a profound understanding of the pattern formation mechanism in each specific system, or complementary, of the universal properties of a class of patterns and their response to symmetry breakings. Common pattern interventions are feed back control [11] and symmetry breaking via spatial, temporal or spatio-temporal modulations [19][20][21][22][23][24][25][26][27][28][29][30]. To mention two interesting scenarios, spatial forcing near resonance (between the forcing and the natural wavelength) can lead to so-called incommensurate patterns [19,20], while symmetry breaking via long-wave spatial modulations can render stationary patterns time dependent [31][32][33][34]. The response of patterns in quasi one-dimensional (1D) systems is now well established, however truely 2D scenarios, like the interplay of an anisotropy and a modulation in different directions, are yet fully unexplored.
In this paper we identify and analyze a new class of quasi-2D anisotropic inhomogeneous pattern forming systems. The wave vector of the patterns lies close to q q , 0 0 0 = (ˆ) along the preferred x-direction (anisotropy), as for the two experimental systems sketched in figure 1: (a) a spatially modulated version of the wrinkle forming system [16,35] and (b) a modulated version of electroconvection (EHC) in nematic liquid crystals [6]. By a modulation we break in such systems the translational symmetry along the perpendicular y-direction, causing a Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
variation of the pattern's natural wave number q . 0 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 addition, however, we find a whole family of stable branched patterns as shown in figure 1(d) and (e). Surprisingly, they have-at identical parameters-different characteristic wave numbers and include different numbers of branching points. Moreover, 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.

Model and generic amplitude equation
A generic model for the formation of stationary periodic patterns in anisotropic-but homogeneous-2D systems, described by a field u x y t , , , ( ) has been proposed in [43]. One interpretation of the field u x y t , , ( )is, that it describes the (small) lateral displacement of a thin elastic plate extended in the x y plane, loaded along the x and y direction and supported by an elastic medium, similar as in wrinkling systems (whereby the in-plane elastic deformations are neglected) [43]. Here we generalize this model to an inhomogeneous situation by modulating the pattern's preferred natural wave number q 0 along the direction perpendicular to the anisotropy (the x-direction), and for different initial conditions by using the pseudo-spectral method [36]. Each picture is a cutout of a larger domain (L q 80 2 with an amplitude M and a wave number k m considerably smaller than q . 0 Then the dynamics of the patterns, described by the real field u x y t , , ( )is governed by The first line corresponds to the original model in [43] and the second line includes contributions due to the modulation q y . 0 ( ) Equation (2) is a representative of the here-identified symmetry class. It can be directly linked to the elastic wrinkle-forming system via an appropriate rescaling [43]: q 0 4 1 = k l relates the critical wave number to the bending stiffness λ 1 of the hard layer and the elastic modulus κ of the substrate. The control parameter Also note that, similar to the homogeneous version [43] and the wrinkle system, equation (2) can be derived from a functional as described in the appendix. For the following, we have chosen W = 1, c = 0.5 and q 1 0 = that favor straight stripe formation parallel to the y-direction for the homogeneous case.
Close to the onset of supercritically bifurcating patterns their generic (system-independent) properties may be described by a nonlinear dynamical equation for the complex envelope A x y t , , , (the star denotes the complex conjugate). This reduction method, the so-called multiple scale analysis, is well established for supercritical bifurcations in 2D homogeneous isotropic [2,18] and anisotropic systems [5,43] or 1D inhomogeneous ones [32]. via the multiple scale analysis from the system described by equation (2) the generic amplitude equation for the here-identified universality class of patterns in anisotropic systems with a spatially varying natural wave number q y : 0 ( )  (3) is exactly the equation for the amplitude of stripe patterns in unmodulated EHC [5,43], the second anisotropic system suggested for experimental investigations of phenomena described here. For all specific pattern forming systems in the considered universality class, equation (3) can be derived via perturbation techniques from the basic equations (as for example from equation (2)) and it has always the same form-only the coefficients will depend on the specific system. The presented results are obtained via the universal equation (3).

Results and discussion
The modulation of the natural wave number, see equation (1) ) we consider only perturbations along the x-direction, as perturbations parallel to the stripes (along y) have been numerically found to have no effect.
For the basic state A = 0 the stability condition,   (L L x y is the domain area) one can hence determine which of the stable patterns is energetically preferred at a given parameter set (see also p. 868 of [2]). We have calculated the energy densities of unbranched patterns f s and of branched patterns f d within the red-colored region in figure 2(b). figure 3(c) as a function of ε and for three different wave numbers Q. For small ε, branched patterns have a lower energy and thus are energetically preferred, whereas for considerably larger ε, unbranched patterns are in turn preferred. The crossover can be understood as the modulation becomes less important at large values of the control parameter ε.
The second central result is that for each parameter set out of a large range beyond Q N e ( )we find always a whole family of stable branched patterns, characterized by different wavelengths and distinct numbers of branching points-and obtained by varying the initial conditions. For example, the two patterns in parts (d) and (e) in figure 1 contain five and seven pairs of branching points at identical parameters. They are both stable, however, the value of the functional of every branched pattern is, in general, different. This pattern coexistence in inhomogeneous systems is a surprising generalization of the well established Eckhaus wave number band of stable stripe patterns in homogeneous systems [37][38][39][40][41][42]. There, stable stripes of different wave numbers coexist at an identical parameter set, as verified in EHC [39], buckling [40] and axisymmetric Taylor vortex flow [41,42].
Here, a band of branched patterns with different wavelengths and numbers of branching points coexists. In both cases, the origin of this coexistence lies in the fact that transitions between two stable periodic patterns require rather high excitations [38] worth to be investigated further.
Another interesting observation is due to the branching points (defects) having opposite topological charges in each half period π/k m along the y-direction hence they repel each other. As shown in figure 4, this repulsion can lead to different orderings of the defects with increasing defect density, for instance, the zig-zag ordering shown in figure 4(b). It can even lead to more complex cascade-like ordering, as shown in figure 4(c). However, how the detailed ordering of branching points can be controlled by the magnitude, the wavelength and the anharmonicity of the modulation is an interesting and fundamental question that needs to be adressed in the future. where the u 4 --term compensates for the cubic term occurring in u t ¶ and similarly the last two terms. These last two terms again vanish due to the periodic boundary condition (here in y-direction):