Ginzburg-Landau type equations are models for superconductivity, superfluidity, Bose-Einstein condensation, etc. A crucial feature is the presence of quantized vortices, which are topological zeroes of the complex-valued solutions. We will present a new result on the derivation of a mean-field limit equation for the dynamics of many vortices starting from the parabolic Ginzburg-Landau equation or the Gross-Pitaevskii (=Schrodinger Ginzburg-Landau) equation.[-]

We propose a platform for finding the global minimum of XY Hamiltonian with polariton graphs. We derive an approximate analytic solution to the spinless complex Ginzburg-Landau equation that describes the density and kinetics of a polariton condensate under incoherent pumping. The analytic expression of the wavefunction is used as the building block for constructing the XY Hamiltonian of two-dimensional polariton graphs. We illustrate examples of the quantum simulator for various classical magnetic phases on some simple lattice geometries: linear, triangular, square.[-]