  We investigate the mobility of nonlinear localized modes in a generalized
discrete Ginzburg-Landau type model, describing a one-dimensional waveguide
array in an active Kerr medium with intrinsic, saturable gain and damping. It
is shown that exponentially localized, traveling discrete dissipative
breather-solitons may exist as stable attractors supported only by intrinsic
properties of the medium, i.e., in absence of any external field or
symmetry-breaking perturbations. Through an interplay by the gain and damping
effects, the moving soliton may overcome the Peierls-Nabarro barrier, present
in the corresponding conservative system, by self-induced time-periodic
oscillations of its power (norm) and energy (Hamiltonian), yielding exponential
decays to zero with different rates in the forward and backward directions. In
certain parameter windows, bistability appears between fast modes with small
oscillations, and slower, large-oscillation modes. The velocities and the
oscillation periods are typically related by lattice commensurability, and
exhibit period-doubling bifurcations to chaotically "walking" modes under
parameter variations. If the model is augmented by inter-site Kerr
nonlinearity, thereby reducing the Peierls-Nabarro barrier of the conservative
system, the existence regime for moving solitons increases considerably, and a
richer scenario appears including Hopf-bifurcations to incommensurately moving
solutions and phase-locking intervals. Stable moving breathers also survive in
presence of weak disorder.
