The defocusing NLS equation with nonzero background: Painlevé asymptotics in two transition regions

In this paper, we address the Painlevé aymptotics in the transition region $\mathrm{\mid }\xi \mathrm{\mid }:=\mid \frac{x}{2t}\mid \approx 1$ to the Cauchy problem of the defocusing Schr$\ddot{\text{o}}$dinger equation with a nonzero background.With the $\bar{\mathrm{\partial }}$-generation of the nonlinear steepest descent approach and double scaling limit to compute the long-time asymptotics of the solution in two transition regions defined as $${\mathcal{P}}_{\pm 1}(x,t):=\{(x,t)\in \mathbb{R}\times {\mathbb{R}}^{+},0<\mathrm{\mid }\xi -(\pm 1)\mathrm{\mid }{t}^{2\mathrm{/}3}\le C\},$$ we find that the long-time asymptotics in both transition regions ${\mathcal{P}}_{\pm 1}(x,t)$ can be expressed in terms of the Painlevé II equation. We are also able to express the leading term explicitly in terms of the Ariy function.