  We investigate the stability and coherence properties of one-dimensional
exciton-polariton condensates under nonresonant pumping. We model the
condensate dynamics using the open-dissipative Gross-Pitaevskii equation. In
the case of spatially homogeneous pumping, we find that the instability of the
steady state leads to significant eduction of the coherence length. We consider
two effects that can lead to the stabilization of the steady state, i.e. the
polariton energy relaxation and the influence of an inhomogeneous pumping
profile. We find that, while the former has little effect on the stability, the
latter is very effective in stabilizing the condensate which results in a large
coherence length.
