The formation of nonlinear Bloch states in open driven-dissipative system of exciton-polaritons loaded into a weak-contrast 1D periodic lattice is studied numerically and analytically. The condensate is described within the framework of mean-field theory by the coupled equations for the order parameter and for the density of incoherent excitons. The stationary nonlinear solutions having the structure of Bloch waves are studied in detail. It is shown that there is a bifurcation leading to the appearance of a family of essentially nonlinear states. The special feature of these solutions is that its current does not vanish when the quasi-momentum of the state approaches the values equal to the half of the lattice constant. To explain the bifurcations found in numerical simulations a simple perturbative approach is developed. The stability of the nonlinear states is examined by linear spectral analysis and by direct numerical simulations. An experimental scheme allowing the observation of the discussed nonlinear current states is suggested and studied by numerical simulations.
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