Travelling-wave amplitudes as solutions of the phase-field crystal equation.

The dynamics of the diffuse interface between liquid and solid states is analysed. The diffuse interface is considered as an envelope of atomic density amplitudes as predicted by the phase-field crystal model (Elder 2004 , 051605 (doi:10.1103/PhysRevE.70.051605); Elder 2007 , 064107 (doi:10.1103/PhysRevB.75.064107)). The propagation of crystalline amplitudes into metastable liquid is described by the hyperbolic equation of an extended Allen-Cahn type (Galenko & Jou 2005 , 046125 (doi:10.1103/PhysRevE.71.046125)) for which the complete set of analytical travelling-wave solutions is obtained by the [Formula: see text] method (Malfliet & Hereman 1996 , 563-568 (doi:10.1088/0031-8949/54/6/003); Wazwaz 2004 , 713-723 (doi:10.1016/S0096-3003(03)00745-8)). The general [Formula: see text] solution of travelling waves is based on the function of hyperbolic tangent. Together with its set of particular solutions, the general [Formula: see text] solution is analysed within an example of specific task about the crystal front invading metastable liquid (Galenko 2015 , 1-10 (doi:10.1016/j.physd.2015.06.002)). The influence of the driving force on the phase-field profile, amplitude velocity and correlation length is investigated for various relaxation times of the gradient flow.This article is part of the theme issue 'From atomistic interfaces to dendritic patterns'.