  We demonstrate that collective motion of interfacial fluctuations can occur
at the interface between two coexisting thermodynamic phases. Based on computer
simulation results for driven diffusive Ising and Blume-Capel models, we
conjecture that the thermal capillary waves at a planar interface travel along
the interface if the lateral order parameter current j_op(y) is an odd function
of the distance y from the interface and hence possesses opposite directions in
the two phases. Such motion does not occur if j_op(y) is an even function of y.
A discrete Gaussian interface model with effective dynamics exhibits similiar
transport phenomena but with a simpler dispersion relation. These findings open
up avenues for controlled interfacial transport on the nanoscale.
