Nonlinear fronts near a first-order phase transition.

Abstract

Arguments are presented for understanding the selection of speed and the nature of the fronts which join stable and unstable states on both the subcritical and the supercritical side of first order phase transitions. Subcritically, a unique front exists for a given set of parameter values, corresponding to a unique connection between metastable states. In the phase space of the Galilean ODE, uniqueness arises from the non-perturbability of a connection between saddle points corresponding to plane waves and ground states. Unique connections are shown to be special solutions which have the Painleve property, and such solutions are found using the WTC method. ODE trajectories corresponding to unique front solutions are shown to satisfy van Saarloos' reduction of order, which suggests that "integrability" for solutions corresponds to the existence of non-generic conservation laws. Supercritically, observable front behavior occurs when the asymptotic spatial structure of a trajectory in the Galilean ODE corresponds to the most unstable temporal mode in the governing PDE. This selection criterion distinguishes between a "nonlinear" front, which originates in the first order nature of the bifurcation, and a "linear" front. The nonlinear front is a continuous deformation of the unique subcritical fronts, is a strongly heteroclinic trajectory in the ODE, and is an integrable solution for the PDE. Many of the characteristics of the linear front are obtained from a steepest descents linear analysis due to Kolmogorov. Its connection with global stability, and in particular with arguments based on a Liapunov functional, is pursued. Liapunov functionals illustrate that stable front behavior arises from the most unstable mode of the PDE, but the stability-based selection criterion is valid even where the Liapunov functional doesn't exist.