THE EXISTENCE OF TRAVELING-WAVE SOLUTIONS OF A GENERALIZED PHASE-FIELD MODEL

This paper establishes the existence and, in certain cases, the unique ness of travelling wave solutions of both second-order and higher-orde r phase-field systems. These solutions describe the propagation of pla nar 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 thick ness. The equations for the transition layer separating the two phases form a system identical to that for the travelling-wave problem, in w hich the temperature is strongly coupled with the order parameter. Thu s 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 t wo second-order equations, we prove a general existence theorem based upon topological methods. A second, constructive proof based upon inva riant-manifold methods is also given when the parameter ct is either s ufficiently small br sufficiently large. In either regime, it is also proved that the wave and the wave velocity are globally unique. Analog ous results are also obtained for generalized phase-field systems in w hich the order parameter solves a higher-order differential equation. In this paper, the higher-order tems occur as a singular peturbation o f the standard (isotropic) second-order equation. The higher-order ter ms are useful in modelling anisotropic interfacial motion.