Fredholm determinant representation of the Painlev\'e II $\tau$-function

TL;DR

This paper expresses the Painlevé II tau-function as a Fredholm determinant of an integrable operator, linking its zeros to the Malgrange divisor and providing a new analytical representation.

Contribution

It introduces a Fredholm determinant formulation of the Painlevé II tau-function, connecting isomonodromic deformations with integrable operator theory.

Findings

Tau-function represented as a Fredholm determinant

Zeros of the tau-function correspond to the Malgrange divisor

Provides a new analytical tool for Painlevé II analysis

Abstract

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