  We study a two-dimensional incoherently pumped exciton-polariton condensate
described by an open-dissipative Gross-Pitaevskii equation for the polariton
dynamics coupled to a rate equation for the exciton density. Adopting a
hydrodynamic approach, we use multiscale expansion methods to derive several
models appearing in the context of shallow water waves with viscosity. In
particular, we derive a Boussinesq/Benney-Luke type equation and its far-field
expansion in terms of Kadomtsev-Petviashvili-I (KP-I) equations for right- and
left-going waves. From the KP-I model, we predict the existence of
vorticity-free, weakly (algebraically) localized two-dimensional dark-lump
solitons. We find that, in the presence of dissipation, dark lumps exhibit a
lifetime three times larger than that of planar dark solitons. Direct numerical
simulations show that dark lumps do exist, and their dissipative dynamics is
well captured by our analytical approximation. It is also shown that lump-like
and vortex-like structures can spontaneously be formed as a result of the
transverse "snaking" instability of dark soliton stripes.
