Absence of Selection in Directional Solidification

We consider a Green's-function formulation of directional solidification. We develop a numerical scheme which suppresses logarithmic singularities and makes the asymptotic behavior explicit. This one is characterized by a new parameter $\lambda$. At zero surface tension, we obtain solutions for any choice of $\alpha$ and $\lambda$. With surface tension, we find that the wavelength remains arbitrary while discrete $\lambda$ values are allowed, in agreement with the work of Dombre and Hakim. Their results are generalized to arbitrary surface tension.