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Fredholm determinant representation of the Painlevé II $τ$-function

Harini Desiraju

Abstract

We formulate the generic $τ$-function of the Painlevé II equation as a Fredholm determinant of an integrable (Its-Izergin-Korepin-Slavnov) operator. The $τ$-function depends on the isomonodromic time $t$ and two Stokes' parameters, and the vanishing locus of the $τ$-function, called the Malgrange divisor is determined by the zeros of the Fredholm determinant.

Fredholm determinant representation of the Painlevé II $τ$-function

Abstract

We formulate the generic -function of the Painlevé II equation as a Fredholm determinant of an integrable (Its-Izergin-Korepin-Slavnov) operator. The -function depends on the isomonodromic time and two Stokes' parameters, and the vanishing locus of the -function, called the Malgrange divisor is determined by the zeros of the Fredholm determinant.

Paper Structure

This paper contains 11 sections, 6 theorems, 125 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Setup
  3. Parametrices
  4. Model problem
  5. Local parametrices
  6. Reduction to a RHP along the imaginary axis
  7. Integrable kernel and Fredholm determinant
  8. LU decomposition
  9. Integrable kernel
  10. Malgrange forms
  11. Proof of theorem \ref{['theorem:1']}

Key Result

Theorem \oldthetheorem

The $\tau$-function of Painlevé II equation can be expressed in terms of a Fredholm determinant of an integrable operator $\widetilde{\mathcal{K}}$ as follows where $\mathcal{F}(t,\nu,h)$ is a regular function of the parameters $t$, $h$ and $\nu$ defined in tdef123, hnu. The kernel of $\widetilde{\mathcal{K}}$ takes the form with the functions (see def:JUMP_ABCD) where $D_{\nu}$ is the paraboli

Figures (6)

  • Figure 1: Stokes rays
  • Figure 2: Deforming the contour in fig. \ref{['fig:1']}
  • Figure 3: Deformed Painlevé II Riemann--Hilbert contour $\Sigma$.
  • Figure 5: Mapping the $\zeta$-plane to the right-half of $z$-plane
  • Figure 6: Reducing the Painlevé II RHP on to the imaginary axis.
  • ...and 1 more figures

Theorems & Definitions (16)

  • Definition 1
  • Theorem \oldthetheorem
  • Remark 1
  • Remark 2
  • Remark 3
  • lemma 1
  • proof
  • proposition 1
  • proof
  • proposition 2
  • ...and 6 more