From Painlev\'e to Zakharov-Shabat and beyond: Fredholm determinants and integro-differential hierarchies

TL;DR

This paper reveals a unifying framework based on Fredholm determinants that connects various integrable systems, including Painlevé hierarchies, KPZ solutions, fermions, and Zakharov-Shabat systems, through a common hierarchy of equations.

Contribution

It introduces a quasi-universal hierarchy of equations that generalizes several integrable models and provides explicit solutions in terms of Fredholm determinants.

Findings

Unified framework for integrable systems using Fredholm determinants

Explicit solution to Zakharov-Shabat inverse scattering in terms of Fredholm determinants

Connection between Painlevé II hierarchy, KPZ, fermions, and Zakharov-Shabat systems

Abstract

As Fredholm determinants are more and more frequent in the context of stochastic integrability, we unveil the existence of a common framework in many integrable systems where they appear. This consists in a quasi-universal hierarchy of equations, partly unifying an integro-differential generalization of the Painlev\'e II hierarchy, the finite-time solutions of the Kardar-Parisi-Zhang equation, multi-critical fermions at finite temperature and a notable solution to the Zakharov-Shabat system associated to the largest real eigenvalue in the real Ginibre ensemble. As a byproduct, we obtain the explicit unique solution to the inverse scattering transform of the Zakharov-Shabat system in terms of a Fredholm determinant.