  Exciton-polaritons generated by light-induced potentials can spontaneously
condense into macroscopic quantum states that display nontrivial spatial and
temporal density modulation. While these patterns and their dynamics can be
reproduced through the solution of the generalized Gross-Pitaevskii equation, a
predictive theory of their thresholds, oscillation frequencies, and
multi-pattern interactions has so far been lacking. Here we represent such an
approach based on the linear non-Hermitian modes of the complex-valued
light-induced potential. We provide a simple analytic expression for the lowest
thresholds that is able to explain the modal patterns observed in recent
experiments for various pump geometries. We also show that the evolution of the
condensate with increasing pump strength is strongly geometry dependent and can
display contrasting features such as enhancement or reduction of the spatial
localization of the condensate.
