  The generalized Gross-Pitaevskii equation (gGPE) is an effective
phenomenological description for the dynamics of incoherently pumped
exciton-polariton condensates. However, a brute force numerical simulation of
the gGPE provides little physical insight into condensate formation under
arbitrary pumping configurations, and is demanding in terms of computational
resources. We introduce in this paper a modal description of polariton
condensation under incoherent pumping of arbitrary spatial profile, based on
eigenmodes of the non-Hermitian generator of the linearized dynamics. A
pump-dependent basis is then introduced to formulate a temporal coupled-mode
theory that captures condensate dynamics in the presence of all nonlinear
interactions. Simulations using a single set of modes for a given pumping and
trapping configuration agree very well with a full integration of the gGPE in
diverse dynamical regimes, supporting the validity of this modal description,
while also providing a speedup in simulation times.
