Traveling wave profiles for a crystalline front invading liquid states: Analytical and numerical solutions

• Manuscript provides analytic solution for traveling wave profiles of the phase field crystal model in two dimensions.
• Solutions are derived for both parabolic and hyperbolic cases for a crystalline state invading both metastable and unstable liquids.
• Solutions are verified numerically and shown to be accurate in the small velocity limit near the liquid/solid phase transition.

The properties of a two dimensional crystalline phase invading a metastable or unstable liquid state are examined using the amplitude expansion formulation of the hyperbolic and parabolic phase-field crystal model. When the amplitudes are real and equal to each other, analytic expressions are derived for the profile of a steady state liquid–solid front traveling at constant velocity. Numerical simulations of the full amplitude formulation are conducted and compared with the analytic results. Close to the melting transition the analytic results for the liquid–solid profile, velocity and width are in quantitative agreement with the numerical results and disagree far from the transition.