Pattern formation in the presence of symmetries

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 quasi-two-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 indefinitely in a well-defined nonequilibrium state. This physical system satisfies, 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 pertubation 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, 1609 (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.