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

2022/11/07 by Zhaoyu Wang, Wang, Zhaoyu, Engui Fan +1
Mathematics · Physics and Astronomy · #35B40 #35C20 #35G25 #35P25 #35Q15 #35Q55 #Advanced Mathematical Physics Problems #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Photonic Systems #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.2211.03914

openalex publication_date 2022/11/07 · openalex created_date 2022/11/14 · openalex updated_date 2026/07/28

Abstract

In this paper, we address the Painlevé aymptotics in the transition region |ξ|:=|(x)/(2t)| ≈ 1 to the Cauchy problem of the defocusing Schrodinger equation with a nonzero background.With the ∂-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 P_{± 1}(x,t):=\ (x,t) ∈ ℝ×ℝ⁺, 0lt;|ξ-(± 1)|t^2/3≤ C\, we find that the long-time asymptotics in both transition regions P_{± 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.

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