Fredholm determinant representation of the homogeneous Painlevé II τ-function
Fredholm determinant representation of the homogeneous Painlevé II τ-function
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 …