  We study numerically the phase diagram and the response under a driving force
of the phase field crystal model for pinned lattice systems introduced recently
for both one and two dimensional systems. The model describes the lattice
system as a continuous density field in the presence of a periodic pinning
potential, allowing for both elastic and plastic deformations of the lattice.
We first present results for phase diagrams of the model in the absence of a
driving force. The nonlinear response to a driving force on an initially pinned
commensurate phase is then studied via overdamped dynamic equations of motion
for different values of mismatch and pinning strengths. For large pinning
strength the driven depinning transitions are continuous, and the sliding
velocity varies with the force from the threshold with power-law exponents in
agreement with analytical predictions. Transverse depinning transitions in the
moving state are also found in two dimensions. Surprisingly, for sufficiently
weak pinning potential we find a discontinuous depinning transition with
hysteresis even in one dimension under overdamped dynamics. We also
characterize structural changes of the system in some detail close to the
depinning transition.
