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The existence of travelling wave solutions of a generalized phase-field model. (English) Zbl 0867.35041

Existence and, in certain cases, uniqueness (sometimes local and sometimes global) of the traveling-wave solutions of heteroclinic types, i.e., describing a steadily propagating phase transition wave, is proved for a system of two coupled nonlinear parabolic, or in a more general case, ultraparabolic equations that describe, in the phase-field approximation, propagation of a solidification front in a hypercooled melt. Two different methods of the proof are used: a more general one, based on topological methods, that provides only existence proof, and a constructive proof based on the center manifold technique, that allows also to prove uniqueness of the heteroclinic traveling wave solution. The rigorous results obtained in the work are corroborated by numerical simulations.

MSC:

35K55 Nonlinear parabolic equations
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
35K70 Ultraparabolic equations, pseudoparabolic equations, etc.
35K45 Initial value problems for second-order parabolic systems
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