Ivantsov parabolic solution for two combined moving interfaces
  We demonstrate that for a migration of a liquid layer between the melting and the solidification front an exact steady-state solution with two parabolic fronts can be found. A necessary condition hereby is that the temperature of the solidification front exceeds the temperature of the melting front (both temperatures are supposed to be constant). It is shown that in pure materials and alloys there exist two types of solutions with two convex and with two concave parabolas respectively. While a steady-state process with two planar interfaces is only possible for a single point, the processes with two parabolas are possible inside a region of control parameters. The relations between the Peclet numbers and the control parameters are obtained. 
D. Temkin