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 (Elderet al. 2004Phys. Rev. E 70 , 051605 (doi:10.1103/PhysRevE.70.051605 ); Elderet al. 2007Phys. Rev. B 75 , 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 2005Phys. Rev. E 71 , 046125 (doi:10.1103/PhysRevE.71.046125 )) for which the complete set of analytical travelling-wave solutions is obtained by themethod (Malfliet & Hereman 1996Phys. Scr. 15 , 563–568 (doi:10.1088/0031-8949/54/6/003 ); Wazwaz 2004Appl. Math. Comput. 154 , 713–723 (doi:10.1016/S0096-3003(03)00745-8 )). The generalsolution of travelling waves is based on the function of hyperbolic tangent. Together with its set of particular solutions, the generalsolution is analysed within an example of specific task about the crystal front invading metastable liquid (Galenkoet al. 2015Phys. D 308 , 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’.