  Directed motion of topological solitons (kinks or antikinks) in the damped
and AC-driven discrete sine-Gordon system is investigated. We show that if the
driving field breaks certain time-space symmetries, the soliton can perform
unidirectional motion. The phenomenon resembles the well known effects of
ratchet transport and nonlinear harmonic mixing. Direction of the motion and
its velocity depends on the shape of the AC drive. Necessary conditions for the
occurrence of the effect are formulated. In comparison with the previously
studied continuum case, the discrete case shows a number of new features:
non-zero depinning threshold for the driving amplitude, locking to the rational
fractions of the driving frequency, and diffusive ratchet motion in the case of
weak intersite coupling.
