Fredholm determinant representation of the Painlev\'e II $\tau$-function
Mathematical Physics
2024-05-01 v4 Classical Analysis and ODEs
math.MP
Exactly Solvable and Integrable Systems
Abstract
We formulate the generic -function of the Painlev\'e 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.
Cite
@article{arxiv.2008.01142,
title = {Fredholm determinant representation of the Painlev\'e II $\tau$-function},
author = {Harini Desiraju},
journal= {arXiv preprint arXiv:2008.01142},
year = {2024}
}
Comments
29 pages, 8 figures. V3: some signs corrected; v4: There was an error with the t-derivative that is fixed in this version. This changes a factor of 2 in the subleasing term of the main result