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Pattern formation in the presence of symmetries. (English) Zbl 1062.76511
Summary: 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 [A. C. Newell and J. A. Whitehead, J. Fluid Mech. 38, 279–303 (1969; Zbl 0187.25102); L. A. Segel, J. Fluid Mech. 38, 203–224 (1969; Zbl 0179.57501)] 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 [J. B. Swift and P. C. Hohenberg, Phys. Rev. A 15, 319 ff (1969)]. Our analysis is complementary to the Cross-Newell approach [M. C. Cross and A. C. Newell, Phys. D 10, 299–328 (1984; Zbl 0592.76052)] 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 Y. Pomeau and P. Manneville [J. Phys. Lett. 40, L609–L612 (1979)]. The theory predicts the existence of stable rhombic arrays with qualitative details that should be system-independent. Our experiments on 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.
MSC:
76E99Hydrodynamic stability
76V05Interacting phases (fluid mechanics)
76M60Symmetry analysis, Lie group and algebra methods (fluid mechanics)