A quantum particle transport induced in a spatially-periodic potential by a
propagating plane wave has a number important implications in a range of
topical physical systems. Examples include acoustically driven semiconductor
superlattices and cold atoms in optical crystal. Here we apply kinetic
description of the directed transport in a superlattice beyond standard linear
approximation, and utilize exact path-integral solutions of the semiclassical
transport equation. We show that the particle drift and average velocities have
non-monotonic dependence on the wave amplitude with several prominent extrema.
Such nontrivial kinetic behaviour is related to global bifurcations developing
with an increase of the wave amplitude. They cause dramatic transformations of
the system phase space and lead to changes of the transport regime. We describe
different types of phase trajectories contributing to the directed transport
and analyse their spectral content
