  We demonstrate that periodic modulation of the nonlinearity coefficient in
the discrete nonlinear Schr\"{o}dinger (DNLS) equation can strongly facilitate
creation of traveling solitons in the lattice. We predict this possibility in
an analytical form, and test it in direct simulations. Systematic simulations
reveal several generic dynamical regimes, depending on the amplitude and
frequency of the time modulation, and on initial thrust which sets the soliton
in motion. These regimes include irregular motion, regular motion of a decaying
soliton, and regular motion of a stable one. The motion may occur in both the
straight and reverse directions, relative to the initial thrust. In the case of
stable motion, extremely long simulations in a lattice with periodic boundary
conditions demonstrate that the soliton keeps moving as long as we can monitor
without any visible loss. Velocities of moving stable solitons are in good
agreement with the analytical prediction, which is based on requiring a
resonance between the ac drive and motion of the soliton through the periodic
potential. All the generic dynamical regimes are mapped in the model's
parameter space. Collisions between moving stable solitons are briefly
investigated too, with a conclusion that two different outcomes are possible:
elastic bounce, or bounce with mass transfer from one soliton to the other. The
model can be realized experimentally in a Bose-Einstein condensate trapped in a
deep optical lattice.
