  We study nonlinear dynamics of exciton-polaritons in an incoherently pumped
semiconductor microcavity with embedded weak-contrast lattice and coupled to an
exciton reservoir. We elucidate fundamental features of non-equilibrium
exciton-polariton condensate trapped in one-dimensional periodical potential
close to zero momentum (so-called "Zero-state") and to the state at the
boundary of Brillouin zone ("$\pi$-state"). Within the framework of the
mean-field theory, we identify different regimes of both relaxation and
oscillatory dynamics of coherent exciton-polaritons governed by superpositions
of Bloch eigenstates within the periodic lattice. In particular, we
theoretically demonstrate stable macroscopical oscillations, akin to nonlinear
Josephson oscillations, between different spectral components of a polariton
condensate in the momenta-space. We elucidate a strong influence of the
dissipative effects and the feedback induced by the inhomogeneity of incoherent
reservoir on the dynamics of the coherent polaritons.
