  We numerically examine driven skyrmions interacting with a periodic quasi-one
dimensional substrate where the driving force is applied either parallel or
perpendicular to the substrate periodicity direction. For perpendicular
driving, the particles in a purely overdamped system simply slide along the
substrate minima; however, for skyrmions where the Magnus force is relevant, we
find that a rich variety of dynamics can arise. In the single skyrmion limit,
the skyrmion motion is locked along the driving or longitudinal direction for
low drives, while at higher drives a transition occurs to a state in which the
skyrmion moves both transverse and longitudinal to the driving direction.
Within the longitudinally locked phase we find a pronounced speed up effect
that occurs when the Magnus force aligns with the external driving force, while
at the transition to transverse and longitudinal motion, the skyrmion velocity
drops, producing negative differential conductivity. For collectively
interacting skyrmion assemblies, the speed up effect is still present and we
observe a number of distinct dynamical phases, including a sliding smectic
phase, a disordered or moving liquid phase, a moving hexatic phase, and a
moving crystal phase. The transitions between the dynamic phases produce
distinct features in the structure of the skyrmion lattice and in the
velocity-force curves. We map these different phases as a function of the ratio
of the Magnus term to the dissipative term, the substrate strength, the
commensurability ratio, and the magnitude of the driving force.
