Fredholm determinant representation of the homogeneous Painlevé II τ-function

Abstract

We formulate the generic τ-function of the homogeneous 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. The vanishing locus of the τ-function, called the Malgrange divisor is then determined by the zeros of the Fredholm determinant.