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

Abstract: In this paper, we address the Painlev\'e aymptotics in the transition region $|\xi|:=\big|\frac{x}{2t}\big| \approx 1$ to the Cauchy problem of the defocusing Schr$\ddot{\text{o}}$dinger equation with a nonzero background.With the $\bar\partial$-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 $$ \mathcal{P}{\pm 1}(x,t):={ (x,t) \in \mathbb{R}\times\mathbb{R}^+, \ \ 0<|\xi-(\pm 1)|t^{2/3}\leq C}, $$ we find that the long-time asymptotics in both transition regions $ \mathcal{P}{\pm 1}(x,t)$ can be expressed in terms of the Painlev\'{e} II equation. We are also able to express the leading term explicitly in terms of the Ariy function.