The existence of travelling wave solutions of a generalized phase-field model

This paper establishes the existence and, in certain cases, the uniqueness of travelling wave solutions of both second-order and higher-order phase-field systems. These solutions describe the propagation of planar solidification fronts into a hypercooled liquid. The equations are scaled in the usual way so that the relaxation time is alpha epsilon(2), where epsilon is a nondimensional measure of the interfacial thickness. The equations for the transition layer separating the two phases form a system identical to that for the travelling-wave problem, in which the temperature is strongly coupled with the order parameter. Thus there is no longer a well-defined temperature at the inteface, as is the case in the more frequently studied situation in which the liquid phase is undercooled but not hypercooled. For phase-held systems of two second-order equations, we prove a general existence theorem based upon topological methods. A second, constructive proof based upon invariant-manifold methods is also given when the parameter ct is either sufficiently small br sufficiently large. In either regime, it is also proved that the wave and the wave velocity are globally unique. Analogous results are also obtained for generalized phase-field systems in which the order parameter solves a higher-order differential equation. In this paper, the higher-order tems occur as a singular peturbation of the standard (isotropic) second-order equation. The higher-order terms are useful in modelling anisotropic interfacial motion.