Traveling waves in rapid solidification

We analyze rigorously the one-dimensional traveling wave problem for a thermodynamically consistent phase field model. Existence is proved for two new cases: one where the undercooling is large but not in the hypercooled regime, and the other for waves which leave behind an unstable state. The qualitative structure of the wave is studied, and under certain restrictions monotonicity of front pro files can be obtained. Further results, such as a bound on propagation velocity and non-existence are discussed. Finally, some numerical examples of monotone and non-monotone waves are provided.