  The coherent nonlinear dynamics of Abrikosov vortices in asymmetric pinning
nanolandscapes is studied by theoretical modeling and combined microwave and dc
electrical resistance measurements. The problem is considered on the basis of a
single-vortex Langevin equation within the framework of a stochastic model of
anisotropic pinning. When the distance over which Abrikosov vortices are driven
during one half ac cycle coincides with one or a multiple of the nanostructure
period, Shapiro steps appear in the current-voltage curves (CVCs) as a general
feature of systems whose evolution in time can be described in terms of a
particle moving in a periodic potential under combined dc and ac stimuli. While
a dc voltage appears in response to the ac drive, the addition of a dc bias
allows one to diminish the rectified voltage and eventually to change its sign
when the extrinsic dc bias-induced asymmetry of the pinning potential starts to
dominate the intrinsic one. This rectified negative voltage in the CVCs becomes
apparent as \emph{zero-bias} Shapiro steps, which are theoretically predicted
and experimentally observed for the first time.
