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We investigate the mobility of nonlinear localized modes in a generalizeddiscrete Ginzburg-Landau type model, describing a one-dimensional waveguidearray in an active Kerr medium with intrinsic, saturable gain and damping. Itis shown that exponentially localized, traveling discrete dissipativebreather-solitons may exist as stable attractors supported only by intrinsicproperties of the medium, i.e., in absence of any external field orsymmetry-breaking perturbations. Through an interplay by the gain and dampingeffects, the moving soliton may overcome the Peierls-Nabarro barrier, presentin the corresponding conservative system, by self-induced time-periodicoscillations of its power (norm) and energy (Hamiltonian), yielding exponentialdecays to zero with different rates in the forward and backward directions. Incertain parameter windows, bistability appears between fast modes with smalloscillations, and slower, large-oscillation modes. The velocities and theoscillation periods are typically related by lattice commensurability, andexhibit period-doubling bifurcations to chaotically "walking" modes underparameter variations. If the model is augmented by inter-site Kerrnonlinearity, thereby reducing the Peierls-Nabarro barrier of the conservativesystem, the existence regime for moving solitons increases considerably, and aricher scenario appears including Hopf-bifurcations to incommensurately movingsolutions and phase-locking intervals. Stable moving breathers also survive inpresence of weak disorder.

Strongly localized moving discrete dissipative breather-solitons in Kerr nonlinear media supported by intrinsic gain