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Asymptotics of Fredholm determinant solutions of the noncommutative Painlevé II equation

Jia-Hao Du, Shuai-Xia Xu, Yu-Qiu Zhao

Abstract

In this paper, we study the asymptotic behavior of a family of pole-free solutions to the noncommutative Painlevé II equation. These particular solutions can be expressed in terms of the Fredholm determinant of the matrix version of the classical Airy operator, which are analogous to the Hastings-McLeod solution and the Ablowitz-Segur solution of the classical Painlevé II equation. Using the Riemann-Hilbert approach, we derive the asymptotics of the Fredholm determinant and the associated particular solutions $β(\vec{s})$ to the noncommutative Painlevé II equation in the regime $\vec{s}=\left(s+\fracτ{\sqrt{-s}},s-\fracτ{\sqrt{-s}}\right)$ with $τ\ge 0$ and $s\to-\infty$. The solutions depend on a two by two Hermitian matrix with eigenvalues in the interval $(-1,1)$. The asymptotics are expressed in terms of one parameter family of special solutions of the classical Painlevé V equation. Furthermore, we derive the asymptotics, including the connection formulas, for this one parameter family of solutions of the Painlevé V equation both as $ix\to -\infty$ and $x\to 0$.

Asymptotics of Fredholm determinant solutions of the noncommutative Painlevé II equation

Abstract

In this paper, we study the asymptotic behavior of a family of pole-free solutions to the noncommutative Painlevé II equation. These particular solutions can be expressed in terms of the Fredholm determinant of the matrix version of the classical Airy operator, which are analogous to the Hastings-McLeod solution and the Ablowitz-Segur solution of the classical Painlevé II equation. Using the Riemann-Hilbert approach, we derive the asymptotics of the Fredholm determinant and the associated particular solutions to the noncommutative Painlevé II equation in the regime with and . The solutions depend on a two by two Hermitian matrix with eigenvalues in the interval . The asymptotics are expressed in terms of one parameter family of special solutions of the classical Painlevé V equation. Furthermore, we derive the asymptotics, including the connection formulas, for this one parameter family of solutions of the Painlevé V equation both as and .

Paper Structure

This paper contains 28 sections, 6 theorems, 214 equations, 1 figure.

Table of Contents

  1. Introduction and statement of results
  2. Asymptotics of one parameter family of solutions of the PV equation
  3. Asymptotics of the Fredholm determinant and the noncommutative PII equation
  4. Model Riemann-Hilbert problem
  5. Model RH problem for $\Psi$
  6. Model RH problem for the PV equation
  7. Vanishing lemma and existence of solution to the model RH problem for $\Phi$
  8. Asymptotic analysis of the RH problem for $\Phi$ as $ix\to -\infty$
  9. Normalization
  10. Opening the lenses and deformation
  11. Global parametrix
  12. Local parametrix at $\lambda=0$
  13. Local parametrix at $\lambda=1$
  14. Final transformation
  15. Proof of Theorem \ref{['thm:2']}: asymptotics of the PV transcendents as $ix\to-\infty$
  16. ...and 13 more sections

Key Result

Theorem 1

Let $\alpha\in i\mathbb{R}$, $\omega\in\mathbb{C}$ and $|\omega|\le 1$. There exists a $\omega$-family of pole-free solutions on the imaginary axis to the Painlevé V system eq:PV for u-eq:PV for y, which satisfies the following asymptotic behavior where and the connection formula between $\mu$ and $\omega$ is given by Moreover, the associated Hamiltonian $H_V(x)$ is pole-free for $x\in i\mathbb

Figures (1)

  • Figure 5: The deformation of the jump contours of the original RH problem.

Theorems & Definitions (19)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Theorem 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Theorem 3
  • Remark 7
  • ...and 9 more