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Painlevé-type asymptotics for the defocusing Manakov system with nonzero boundary conditions

Haibing Zhang, Xianguo Geng, Ruomeng Li, Huan Liu

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 Painlevé II equation. The proof is rigorously established by applying the Deift-Zhou nonlinear steepest descent method to the associated Riemann Hilbert problem.

Painlevé-type asymptotics for the defocusing Manakov system with nonzero boundary conditions

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 matrix Riemann Hilbert problem. We find that they exhibit interesting asymptotic behavior within a narrow transition zone in the - 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 Painlevé II equation. The proof is rigorously established by applying the Deift-Zhou nonlinear steepest descent method to the associated Riemann Hilbert problem.
Paper Structure (17 sections, 24 theorems, 296 equations, 7 figures)

This paper contains 17 sections, 24 theorems, 296 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Statement of results
  3. Analysis of the model RH problem
  4. Definition of the model problem
  5. Proof of Theorem \ref{['Th:asXi']}
  6. Review of the IST for the defocusing Manakov system with NZBCs
  7. Long time asymptotics
  8. The first transformation
  9. The second transformation
  10. The third transformation
  11. The local parametrix
  12. Final transformation and the small norm RH problem
  13. Proof of Theorem \ref{['Th:main']}
  14. Concluding Remarks
  15. Proof of Lemma \ref{['Th:rel']}
  16. ...and 2 more sections

Key Result

Theorem 1.4

Suppose that there exists a global solution $\mathbf{q}(x,t)$ to the defocusing Manakov system E:demanakovS with the NZBCs E:bjtj and its initial data satisfies Assumptions As:gesa and As:1. Let the functions $\delta_1(z)$, $\delta^{\sharp}(z)$ and $\mathrm{P}_1(z)$ be defined by E:th1, E:th2 and E: where the parameter $y$ is given by and $u_{HM}(y)$ is the Hastings--McLeod solution of Painlevé I

Figures (7)

  • Figure 1: The jump contour $\mathcal{X}$ for the model problem $\boldsymbol{N}^{\mathcal{X}}$.
  • Figure 2: From left to right: The signature tables for $\phi_{21}$, $\phi_{32}$ and $\phi_{31}$ for $\xi=-1$ and $q_0=1$. The purple regions correspond to $\{z: \mathrm{Re} \phi_{ij}<0 \}$ and the pink regions to $\{z: \mathrm{Re} \phi_{ij}>0 \}$.
  • Figure 3: The green curve denotes $\mathcal{L}$, and the dashed lines $\mathcal{L}_1$ and $\mathcal{L}_2$ represent the tangent lines at the point $-q_0$.
  • Figure 4: The contour $\Sigma^{(1)}$ (solid), and the region $\{ D_j\}_{j=1}^9$. The dashed line corresponds to the contour where $\Re \Phi_{31}(\xi,z) =0$.
  • Figure 5: The contour $\mathcal{X}^{\epsilon}$ is the solid line within the dashed circle; the region enclosed by the dashed circle is $\mathcal{D}_{\epsilon}$.
  • ...and 2 more figures

Theorems & Definitions (53)

  • Definition 1.1
  • Theorem 1.4: Painlevé asymptotics in $\mathcal{P}_L$
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7: On another Painlevé region
  • Theorem 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 43 more