Some Aspects of the Weakly Non-linear Theory of the Morphological Instability

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Abstract
In this paper we consider the weakly non-linear spatial and temporal evolution of the morphological instability. We conduct our analysis using the method of multiple scales to derive an asymptotic solution for the system near the critical state. For the case of interface breakdown to rolls we derive an amplitude equation with a cubic non-linearity which is the same as that obtained by Newell & Whitehead [J. Fluid Mech. (1969) 38, 279–303] in the context of Bénard convection. For the case of breakdown to a hexagonal structure we derive an amplitude equation with a quadratic non-linearity. This represents a generalization and modification of that proposed by Sriranganathan, Wollkind & Oulton [J. Cryst Growth (1983) 62, 265–283].

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