  We consider the existence, stability and dynamics of the ground state and
nonlinear excitations, in the form of dark solitons, for a
quasi-one-dimensional polariton condensate in the presence of pumping and
nonlinear damping. We find a series of remarkable features that can be directly
contrasted to the case of the typically energy-conserving ultracold alkali-atom
Bose-Einstein condensates. For some sizeable parameter ranges, the nodeless
("ground") state becomes {\it unstable} towards the formation of {\em stable}
nonlinear single or {\em multi} dark-soliton excitations. It is also observed
that for suitable parametric choices, the instability of single dark solitons
can nucleate multi-dark-soliton states. Also, for other parametric regions,
{\em stable asymmetric} sawtooth-like solutions exist. Finally, we consider the
dragging of a defect through the condensate and the interference of two
initially separated condensates, both of which are capable of nucleating dark
multi-soliton dynamical states.
