  We predict the structural interaction of crystalline solid-melt interfaces
using amplitude equations which are derived from classical density functional
theory or phase-field-crystal modeling. The solid ordering decays exponentially
on the scale of the interface thickness at solid-melt interfaces; the overlap
of two such profiles leads to a short range interaction, which is mainly
carried by the longest-range density waves, which can facilitate grain boundary
premelting. We calculate the tail of these interactions, depending on the
relative translation of the two crystals fully analytically and predict the
interaction potential, and compare it to numerical simulations. For grain
boundaries the interaction is predicted to decay twice faster as for two
crystals without misorientation.
