  We present a comprehensive analytical theory of localized nonlinear
excitations - dark solitons, supported by an incoherently pumped, spatially
homogeneous exciton-polariton condensate. We show that, in contrast to dark
solitons in conservative systems, these nonlinear excitations "relax" by
blending with the background at a finite time, which critically depends on the
parameters of the condensate. Our analytical results for trajectory and
lifetime are in excellent agreement with direct numerical simulations of the
open-dissipative mean-field model. In addition, we show that transverse
instability of quasi-one-dimensional dark stripes in a two-dimensional
open-dissipative condensate demonstrates features that are entirely absent in
conservative systems, as creation of vortex-antivortex pairs competes with the
soliton relaxation process.
