Convergence of nonlinear Schrodinger-Poisson systems to the compressible Euler equations

The combined semi-classical and quasineutral limit in the bipolar defocusing nonlinear Schrodinger-Poisson system in the whole space is proven. The electron and current densities, defined by the solution of the Schrodinger Poisson system, converge to the solution of the compressible Euler equation with nonlinear pressure. The corresponding Wigner function of the Schrodinger Poisson system converges to a solution of a nonlinear Vlasov equation. The proof of these results is based On estimates of a modulated energy functional and on the Wigner measure method.