Dynamics of curved fronts and pattern selection.

The stability of moving curved fronts to short wavelength perturbations is investigated using the WKB approximation. A discrete spectrum of unstable localized modes is found for a one parameter family of interfaces. An approximate "selection criterion" for determining the values of the parameter corresponding to the steady state is proposed. For the Saffman-Taylor problem and dendritic growth we prove the conjecture that the n'th selected state possesses n unstable tip-splitting modes.