  In spite of having finite life-time exciton-polaritons in microcavities are
known to condense at strong enough pumping of the reservoir. We present an
analytical theory of such Bose-condensates on a set of localized one-particle
states: condensation centers. To understand physics of these arrays one has to
supplement the Josephson coupling by the radiative coupling caused by the
interference of the light emitted by different centers. Combination of these
couplings with the one-site interaction between the bosons leads to a rich
nonlinear dynamics. In particular, a new regime of radiation appears. We call
it weak lasing: the centers have macroscopic occupations and radiate
coherently, but the coupling alone is sufficient for stabilization. The system
can have several stable states and switch between them. Moreover, the time
reversal symmetry in this regime is, as a rule, broken. A number of existing
experimental puzzles find natural interpretation in the framework of this
theory.
