Resonant interactions and traveling-solidification cells

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Abstract
Recently a set of new phenomena has been observed in the directional solidification of liquid crystals, which include solitary inclusions of traveling waves and spatial period doubling. We propose that these patterns can be naturally explained as arising from the resonant interaction of modes with wave vectors q and 2q. This picture is substantiated by numerical calculations employing the material parameters of the liquid-crystal system of Simon, Bechhoefer, and Libchaber [Phys. Rev. Lett. 61, 2557 (1988)]. Our approach correctly predicts period-doubled (mixed) modes and the parity-breaking bifurcation to the traveling-wave state. Possible causes for the disagreement of some stability assignments as compared with the experiments are sketched.