Simple model for dynamical melting of moving vortex lattices interacting with periodic pinning

Melting of moving vortex lattices in clean superconducting films with periodic pinning is studied by a dynamical ``cage'' model, based on a mean-field treatment of Langevin equation for the whole vortex lattice, assuming elastic flow and that the lattice is incommensurate with the periodic pinning. In the frame moving with the velocity of the vortex lattice center of mass, the model describes forced vibrations of a single vortex tied to a spring. The vortex displacements due to thermal fluctuations and to the periodic pinning force, and the relationship between the driving force and the vortex velocity $(V\ensuremath{-}I$ curves) are obtained by a perturbation method, valid for high velocities and for both weak and strong periodic pinning. The dynamical melting temperature is calculated as a function of the vortex velocity using Lindemann criterion. Application to a square defect lattice gives, for strong pinning, dynamical melting lines and anisotropic $V\ensuremath{-}I$ curves that agree qualitatively with numerical results.