From Painlevé to Zakharov–Shabat and beyond: Fredholm determinants and integro-differential hierarchies. (29th December 2020)

• Record Type: Journal Article
• Authors: Krajenbrink, Alexandre
• Is Part Of: Journal of physics. Volume 54:Number 3(2021)
• Publication Date: 2020-12-29
• Subjects: Painlevé transcendent -- Zakharov–Shabat system -- Riemann–Hilbert problem -- Kardar–Parisi–Zhang equation -- Fredholm determinant; Mathematical physics -- Periodicals; Statistical physics -- Periodicals; Quantum theory -- Periodicals; Matter -- Properties -- Periodicals; 530.105
• DOI: 10.1088/1751-8121/abd078
• Languages: English
• ISSNs: 1751-8113

Abstract

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 Painlevé 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.