  The family of one-dimensional localized solutions to dissipative nonlinear
equations includes a variety of objects such as sources, sinks, shocks (kinks),
and pulses. These states are in general accompanied by nontrivial density
currents, which are not necessarily related to the movement of the object
itself. We investigate the existence and physical properties of sink-type
solutions in nonresonantly pumped exciton-polariton condensates modeled by an
open-dissipative Gross-Pitaevskii equation. While sinks possess density
profiles similar to bright solitons, they are qualitatively different objects
as they exist in the case of repulsive interactions and represent a
heteroclinic solution. We show that sinks can be created in realistic systems
with appropriately designed pumping profiles. We also conclude that in
two-dimensional configurations, due to the proliferation of vortices, sinks do
not appear.
