Parity breaking in directional solidification: Numerics versus amplitude equations

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Abstract
Motivated by recent experiments on the directional solidification of nematic liquid crystals, the standard solidification model is investigated in detail, numerically (by the boundary-integral method) as well as analytically (using amplitude equations). For the parameter values suitable for the experiments, both approaches yield a parity-breaking instability to traveling waves as well as a period-doubling bifurcation. Moreover, they agree with each other quantitatively not only in the immediate vicinity of the expansion point of the amplitude equations but over quite some range of parameters. The resulting phase diagram is in general accord with that obtained in the experiments. Since the amplitude equations describe the interaction of two modes with resonating wave numbers q and 2q, this demonstrates that both experimentally observed transitions have the same origin: the 1:2-mode interaction. In addition, a third oscillatory instability observed experimentally can also be traced back to this interaction. Some questions concerning the direction of some bifurcations are, however, as yet unresolved. Possible causes for these discrepancies are indicated. For other parameter values the amplitude equations predict that no traveling waves will arise from this mode interaction.