On directional solidification of a faceted crystal

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Abstract
We study the Mullins-Sekerka instability of a binary mixture submitted to directional solidification in the case where the basic planar solid-liquid interface is a facet. We show that, for pulling velocities larger than the standard MS instability threshold, there exist a continuum of stationary non-planar periodic front profiles of small amplitude, consisting of an alternation of hot and cold facets connected by curved regions. These solutions are unstable as regards their amplitude, so that they may be viewed as an amplitude threshold for the usual cellular instability, which should therefore exhibit an anomalously large hysteresis