  Using numerical simulations, we examine the nonlinear dynamics of skyrmions
driven over random pinning. For weak pinning, the skyrmions depin elastically,
retaining sixfold ordering; however, at the onset of motion there is a dip in
the magnitude of the structure factor peaks due to a decrease in positional
ordering, indicating that the depinning transition can be detected using the
structure factor even within the elastic depinning regime. At higher drives the
moving skyrmion lattice regains full ordering. For increasing pinning strength,
we find a transition from elastic to plastic depinning that is accompanied by a
sharp increase in the depinning threshold due to the proliferation of
topological defects, similar to the peak effect found at the elastic to plastic
depinning transition in superconducting vortex systems. For strong pinning and
strong Magnus force, the skyrmions in the moving phase can form a strongly
clustered or phase separated state with highly modulated skyrmion density,
similar to that recently observed in continuum-based simulations for strong
disorder. As the Magnus force is decreased, the density phase separated state
crosses over to a dynamically phase separated state with uniform density but
with flow localized in bands of motion, while in the strongly damped limit,
both types of phase separated states are lost. In the strong pinning limit, we
find highly nonlinear velocity-force curves in the transverse and longitudinal
directions, along with distinct regions of negative differential conductivity
in the plastic flow regime. The negative differential conductivity is absent in
the overdamped limit. The Magnus force is responsible for both the negative
differential conductivity and the clustering effects, since it causes faster
moving skyrmions to partially rotate around slower moving or pinned skyrmions.
