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

Xianguo Geng, Haibing Zhang, Jiao Wei

TL;DR

This work establishes the long-time asymptotics for the defocusing Manakov system on a nonzero background by formulating a 3×3 Riemann–Hilbert problem and applying a nonlinear steepest-descent analysis. The authors develop a sequence of contour and function-transformations to reduce the problem to a small-norm RH problem, revealing a leading modulated multisoliton term plus a vector-specific dispersive correction of order t^{-1/2} in the soliton region, with precise error bounds. They construct an outer parametrix built from a modulated pure-soliton RH problem and model inner problems near critical points, yielding explicit radiation terms and a complete reconstruction of q(x,t). The results demonstrate the extra dispersive correction in the vector case compared to the scalar NZBC problem and lay groundwork for extensions to N-component systems and transition-region Painlevé-type behavior.

Abstract

The Manakov system is a two-component nonlinear Schrödinger 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.

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

TL;DR

This work establishes the long-time asymptotics for the defocusing Manakov system on a nonzero background by formulating a 3×3 Riemann–Hilbert problem and applying a nonlinear steepest-descent analysis. The authors develop a sequence of contour and function-transformations to reduce the problem to a small-norm RH problem, revealing a leading modulated multisoliton term plus a vector-specific dispersive correction of order t^{-1/2} in the soliton region, with precise error bounds. They construct an outer parametrix built from a modulated pure-soliton RH problem and model inner problems near critical points, yielding explicit radiation terms and a complete reconstruction of q(x,t). The results demonstrate the extra dispersive correction in the vector case compared to the scalar NZBC problem and lay groundwork for extensions to N-component systems and transition-region Painlevé-type behavior.

Abstract

The Manakov system is a two-component nonlinear Schrödinger 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 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 , but this term does not exist in the scalar case, and we provide the explicit expression for this dispersion term.
Paper Structure (29 sections, 28 theorems, 295 equations, 12 figures)

This paper contains 29 sections, 28 theorems, 295 equations, 12 figures.

Table of Contents

  1. Introduction
  2. The preparation
  3. Main results
  4. A Riemann-Hilbert formulation
  5. Long-time asymptotics in $\mathcal{R}^{+}_{sol}$
  6. The transformation: $\mathbf{M} \to \mathbf{M}^{(1)}$
  7. The transformation: $\mathbf{M}^{(1)} \to \mathbf{M}^{(2)}$
  8. The transformations: $\mathbf{M}^{(2)} \to \mathbf{M}^{(3)} \to \mathbf{M}^{(4)}$
  9. The transformation: $\mathbf{M}^{(4)} \to \mathbf{M}^{(5)}$
  10. The transformations: $\mathbf{M}^{(5)} \to \mathbf{M}^{(6)} \to \mathbf{M}^{(7)}$
  11. The outer parametrix and the local parametrix
  12. Final Transformation: $\mathbf{M}^{(7)} \to \boldsymbol{\mathcal{E}}$
  13. Proof of the asymptotic formula \ref{['E:asyofq']}
  14. Long-time asymptotics in $\mathcal{R}^{-}_{sol}$
  15. The transformations: $\mathbf{M} \to \mathbf{M}^{(1)} \to \mathbf{M}^{(2)} \to \mathbf{M}^{(3)}$
  16. ...and 14 more sections

Key Result

Theorem 1.6

Suppose $\mathbf{q}: {\mathbb R} \times [0,\infty) \to \mathbb{C}^2$ is a smooth solution of the defocusing Manakov system E:demanakovS with the NZBCs E:bjtj, and its initial value $\mathbf{q}_0(x)$ satisfies Assumptions As:1. Let the functions $\delta_1(z)$, $\delta(z)$ and $\delta^{\sharp}(z)$ be where $\mathbf{q}^{[N]}_{msol}(x,t)$ is given by E:msol, which is the modulated $N$-soliton. The co

Figures (12)

  • Figure 1: From left to right: The signature tables for $\phi_{32}$, $\phi_{21}$ and $\phi_{31}$ for $\xi=0.5$ and $q_0=1$. The grey regions correspond to $\{z: \mathrm{Re} \phi_{ij}<0 \}$ and the white regions to $\{z: \mathrm{Re} \phi_{ij}>0 \}$.
  • Figure 2: The contour $\Sigma^{(1)}$ and the regions $\{\mathcal{R}_j\}_{j=1}^2$.
  • Figure 3: The contour $\Sigma^{(3)}$ and regions $\{\Omega_j \}_{j=1}^2$.
  • Figure 4: The contour $\Sigma^{(4)}$ and regions $\Omega_{\pm}$.
  • Figure 5: The contour $\Sigma^{(5)}$ and the regions $\{ D_{j} \}_{j=1}^6$.
  • ...and 7 more figures

Theorems & Definitions (58)

  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6: Asymptotics in the soliton region $\mathcal{R}_{sol}$
  • proof
  • Theorem 1.7
  • proof
  • Remark 1.8
  • Remark 1.9
  • ...and 48 more