Painlev\'e-type asymptotics for the defocusing Manakov system with nonzero boundary conditions

TL;DR

This paper analyzes the long-time asymptotics of solutions to the defocusing Manakov system with nonzero boundary conditions, revealing a transition zone where solutions relate to Painlevé II solutions, using Riemann-Hilbert problem techniques.

Contribution

It provides the first rigorous derivation of Painlevé-type asymptotics for the defocusing Manakov system with nonzero boundaries, employing the Deift-Zhou steepest descent method.

Findings

Asymptotic behavior characterized within a narrow transition zone.

Leading-order term expressed via Painlevé II Hastings-McLeod solution.

Error bounds established for the asymptotic approximation.

Abstract

We investigate the long-time asymptotic behavior of a class of solutions to the defocusing Manakov system under nonzero boundary conditions. These solutions are characterized by a $3 \times 3$ matrix Riemann Hilbert problem. We find that they exhibit interesting asymptotic behavior within a narrow transition zone in the $x$-$t$ plane. We determine the leading-order asymptotic term and the error bound in this region, and we demonstrate that the leading term can be expressed in terms of the Hastings-McLeod solution of the Painlev\'e II equation. The proof is rigorously established by applying the Deift-Zhou nonlinear steepest descent method to the associated Riemann Hilbert problem.