Pattern formation in the presence of symmetries

Abstract

We present a detailed theoretical study of pattern formation in planar continua with translational, rotational, and reflection symmetry. The theoretical predictions are tested in experiments on a quasitwo-dimensional reaction-diffusion system. Spatial patterns form in a chlorite-iodide-malonic acid reaction in a thin gel layer reactor that is sandwiched between two continuously refreshed reservoirs of reagents; thus, the system can be maintained inde6nitely in a well-defined nonequilibrium state. This physical system satis6es, to a very good approximation, the Euclidean symmetries assumed in the theory. The theoretical analysis, developed in the amplitude equation formalism, is a spatiotemporal extension of the normal form. The analysis is identical to the Newell-Whitehead-Segel theory [J. Fluid Mech. 38, 203 (1969); 38, 279 (1969)] at the lowest order in perturbation, but has the advantage that it exactly preserves the Euclidean symmetries of the physical system. Our equations can be derived by a suitable modification of the perturbation expansion, as shown for two variations of the Swift-Hohenberg equation [Phys. Rev. A 15, 319 (1977)]. Our analysis is complementary to the Cross-Newell approach [Physica D 10, 299 (1984)] to the study of pattern formation and is equivalent to it in the common domain of applicability. Our analysis yields a rotationally invariant generalization of the phase equation of Pomeau and Manneville [J. Phys. Lett. 40, L609 (1979)]. The theory predicts the existence of stable rhombic arrays with qualitative details that should be system independent. Our experiments in the reaction-diffusion system yield patterns in good accord with the predictions. Finally, we consider consequences of resonances between the basic modes of a hexagonal pattern and compare the results of the analysis with experiments.

Structural short-range forces between solid-melt interfaces

“…(42) and (43). This expression is valid as long as the overlap of the density waves is still small, such that the nonlinear energy contributions can be neglected.…”

On the projection of functions invariant under the action of a crystallographic group

“…Knowledge of the symmetries is thus an essential step in dealing with this problem. For a more detailed treatment we refer the reader to [9,Chapter 5] and [11] where the 2-dimensional case is discussed, to [7] and, for issues arising from a projection, to [10].…”

“…Models for patterns in thin three-dimensional layers often make the simplifying assumption that the domain is two-dimensional (see, for instance Gunaratne et al [11]). In many cases the simplified model gives a good reproduction of experimental results, as in those discussed by Golubitsky and Stewart [9,Chapter 5].…”

Phase-field-crystal study of grain boundary premelting and shearing in bcc iron

“…To gain additional insights into the role of this parameter, we also compute the disjoining potential using amplitude equations [56][57][58][59][60][61][62][63] . These equations can be formally derived from the PFC model in the small ǫ limit using similar multiscale expansions introduced previously to analyze continuum models of pattern formation [64][65][66][67][68][69][70][71] . While amplitude equations have recently been shown to break down for high angle GBs because of issues related to frame invariance 63 , they are asymptotically exact for small ǫ and low angle GBs.…”