  Open dissipative systems of quantum fluids have been well studied
numerically. In view of a complementary analytical description we extend here
the variational optimization method for Bose-Einstein condensates of closed
systems to open-dissipative condensates. The resulting projection optimization
method is applied to a complex Gross-Pitaevski equation, which models
phenomenologically a photon Bose-Einstein condensate. Together with known
methods from hydrodynamics we obtain an approximate vortex solution, which
depends on the respective open system parameters and has the same properties as
obtained numerically in the literature.
