  We numerically study the behavior of two-dimensional skyrmions in the
presence of a quasi-one-dimensional sinusoidal substrate under the influence of
externally applied dc and ac drives. In the overdamped limit, when both dc and
ac drives are aligned in the longitudinal direction parallel to the direction
of the substrate modulation, the velocity-force curves exhibit classic Shapiro
step features when the frequency of the ac drive matches the washboard
frequency that is dynamically generated by the motion of the skyrmions over the
substrate, similar to previous observations in superconducting vortex systems.
In the case of skyrmions, the additional contribution to the skyrmion motion
from a non-dissipative Magnus force shifts the location of the locking steps to
higher dc drives, and we find that the skyrmions move at an angle with respect
to the direction of the dc drive. For a longitudinal dc drive and a
perpendicular or transverse ac drive, the overdamped system exhibits no Shapiro
steps; however, when a finite Magnus force is present we find pronounced
transverse Shapiro steps along with complex two-dimensional periodic orbits of
the skyrmions in the phase-locked regimes. Both the longitudinal and transverse
ac drives produce locking steps whose widths oscillate with increasing ac drive
amplitude. We examine the role of collective skyrmion interactions and find
that additional fractional locking steps occur for both longitudinal and
transverse ac drives. At higher skyrmion densities, the system undergoes a
series of dynamical order-disorder transitions, with the skyrmions forming a
moving solid on the phase locking steps and a fluctuating dynamical liquid in
regimes between the steps.
