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

Abstract

In this paper, we address the Painlevé aymptotics in the transition region $|\hat{I}¾|:=|\frac{x}{2t}|\approx 1$ 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 $${\mathcal{P}}_{\pm 1}(x,t):=\{(x,t)\in \mathbb{R}\times {\mathbb{R}}^{+},0<|\hat{I}¾-(\pm 1)|t^{2/3}\le C\},$$ 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.

51 pages