Linear stability of directional solidification cells

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Abstract
We formulate the problem of finding the stability spectrum of the cellular pattern seen in directional solidification. This leads to a nonlinear eigenvalue problem for an integro-differential operator. We solve this problem numerically and compare our results to those obtained by linearizing the eigenvalue problem by employing the quasistatic approximation. Contrary to some recent claims, we find no evidence for a Hopf bifurcation to a dendritic pattern.