Defocusing NLS equation with a nonzero background: Painleve asymptotics in two transition regions

We address the Painleve asymptotics of the solution in two transition regions for the defocusing nonlinear Schrodinger (NLS) equation with finite density initial data. The key to prove this result is the formulation and analysis of a Riemann-Hilbert problem associated with the Cauchy problem for the defocusing NLS equation. With the Dbar generalization of the Deift-Zhou nonlinear steepest descent method and double scaling limit technique, in two transition regions, we find that the leading order approximation to the solution of the defocusing NLS equation can be expressed in terms of the solution of the Painleve II equation.