  We theoretically investigate the time dependence of the first order coherence
function for a one-dimensional driven dissipative non-equilibrium condensate.
Simulations on the generalized Gross-Pitaevskii equation (GGPE) show that the
characteristic time scale of exponential decay agrees with the linearized
Bogoliubov theory in the regime of large interaction energy. For very weak
interactions, the temporal correlation deviates from the linear theory, and
instead respects the dynamic scaling of the Kardar-Parisi-Zhang universality
class. This nonlinear dynamics is found to be quantitatively captured by a
noisy Kuramoto-Sivashinsky equation for the phase dynamics.
