The defocusing NLS equation with nonzero background: Painlev\'e asymptotics in two transition regions

TL;DR

This paper analyzes the long-time asymptotics of the defocusing nonlinear Schrödinger equation with nonzero background near critical transition regions, revealing solutions expressed through Painlevé II equations and explicit Airy function formulas.

Contribution

It introduces a novel application of the $ar{ ext{d}}$-steepest descent method and double scaling limits to derive Painlevé II asymptotics in transition regions for the defocusing NLS with nonzero background.

Findings

Asymptotics in transition regions are expressed via Painlevé II equations.

Explicit formulas for leading terms involve Airy functions.

Results extend understanding of long-time behavior in nonzero background scenarios.

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.