Large-time asymptotics for the defocusing Manakov system on nonzero background

TL;DR

This paper derives the long-time asymptotics for the defocusing Manakov system on nonzero background, revealing a modulated multisoliton structure and a unique dispersive correction term absent in scalar cases.

Contribution

It provides the first detailed derivation of the long-time asymptotics for the defocusing Manakov system with nonzero boundary conditions using Riemann-Hilbert analysis.

Findings

Solution exhibits a modulated multisoliton structure

Dispersive correction term of order t^{-1/2} identified

Explicit expression for the dispersion term provided

Abstract

The Manakov system is a two-component nonlinear Schr\"odinger equation. The long-time asymptotics for the defocusing or focusing Manakov system under nonzero background still remains open. In this paper, we derive the long-time asymptotic formula for the solution of the defocusing Manakov system on nonzero boundary conditions and provide a detailed proof. The solution of the defocusing Manakov system on such nonzero background is first transformed into the solution of a $3 \times 3$ matrix Riemann-Hilbert problem. Then we demonstrate how to conduct the Deift-Zhou steepest descent analysis for this Riemann-Hilbert problem, thereby obtaining the long-time asymptotic behavior of the solution in the space-time soliton region. In this region, the leading order of the solution takes the form of a modulated multisoliton. Apart from the error term, we also discover that the defocusing Manakov system has a dispersive correction term of order $t^{-1/2}$, but this term does not exist in the scalar case, and we provide the explicit expression for this dispersion term.