A Riemann-Hilbert approach to Fredholm determinants of Hankel composition operators: scalar-valued kernels

Abstract

We characterize Fredholm determinants of a class of Hankel composition operators via matrixvalued Riemann-Hilbert problems, for additive and multiplicative compositions. The scalar-valued kernels of the underlying integral operators are not assumed to display the integrable structure known from the seminal work of Its, Izergin, Korepin and Slavnov [41]. Yet we are able to describe the corresponding Fredholm determinants through a naturally associated Riemann-Hilbert problem of Zakharov-Shabat type by solely exploiting the kernels' Hankel composition structures. We showcase the efficiency of this approach through a series of examples, we then compute several rank one perturbed determinants in terms of Riemann-Hilbert data and finally derive Akhiezer-Kac asymptotic theorems for suitable kernel classes.

Soliton versus the gas: Fredholm determinants, analysis, and the rapid oscillations behind the kinetic equation

“…The expression (2.3) for the mKdV soliton gas solution is strikingly similar to the Tracy-Widom Fredholm determinant formula [48] for the solution of the integrated version of the defocusing mKdV: 𝑞 𝑡 − 𝑞 𝑥𝑥𝑥 + 6𝑞 2 𝑞 𝑥 = 0. This expression is also similar to the one considered in [4] and in [33] for solving the weak noise theory of the Kardar-Parisi-Zhang equation and more generally in the study of Fredholm determinants of a class of Hankel composition operators [10]. For example…”

Integrable operators, ∂ ― -problems, KP and NLS hierarchy

“…[22,23] for initial data vanishing at infinity and [25] for step-like initial data). In a more general setting such operators can be reduced to integrable operators via Fourier transform, see for example the works [5,10,39]. Applications of this class of operators to the theory of random matrices, integrable probability and integro-differential Painlevé equations and non commutative Painlevé equations are obtained in [2, 11-13, 15, 16, 49].…”