From Painlevé to Zakharov-Shabat and beyond: Fredholm determinants and integro-differential hierarchies

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.

41 pages

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Cited by in corpus (11)

• The inverse scattering of the Zakharov-Shabat system solves the weak noise theory of the Kardar-Parisi-Zhang equation
• Inverse scattering solution of the weak noise theory of the Kardar-Parisi-Zhang equation with flat and Brownian initial conditions
• Airy kernel determinant solutions to the KdV equation and integro-differential Painlevé equations
• Uniform tail asymptotics for Airy kernel determinant solutions to KdV and for the narrow wedge solution to KPZ
• Unitary matrix models and random partitions: Universality and multi-criticality
• Half-space stationary Kardar-Parisi-Zhang equation beyond the Brownian case
• Integrability in the weak noise theory
• Universality for multiplicative statistics of Hermitian random matrices and the integro-differential Painlevé II equation
• Asymptotic expansions for a class of Fredholm Pfaffians and interacting particle systems
• Jánossy densities and Darboux transformations for the Stark and cylindrical KdV equations
• Multicritical Schur measures and higher-order analogues of the Tracy-Widom distribution