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

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.