Integrable operators, $\overline{\partial}$-Problems, KP and NLS hierarchy

TL;DR

This paper develops a theory connecting integrable operators on complex domains with $ar{ ext{}}$-problems, showing how their determinants relate to KP and NLS hierarchies through isomonodromic deformations.

Contribution

It introduces a new framework for integrable operators on complex domains, linking their resolvent and determinants to $ar{ ext{}}$-problems and hierarchies like KP and NLS.

Findings

Resolvent operators derived from $ar{ ext{}}$-problems.

Malgrange one form is closed and relates to the determinant.

Hilbert-Carleman determinant acts as a $ au$-function for KP and NLS.

Abstract

We develop the theory of integrable operators $\mathcal{K}$ acting on a domain of the complex plane with smooth boundary in analogy with the theory of integrable operators acting on contours of the complex plane. We show how the resolvent operator is obtained from the solution of a $\overline{\partial}$-problem in the complex plane. When such a $\overline{\partial}$-problem depends on auxiliary parameters we define its Malgrange one form in analogy with the theory of isomonodromic problems. We show that the Malgrange one form is closed and coincides with the exterior logarithmic differential of the Hilbert-Carleman determinant of the operator $\mathcal{K}$. With suitable choices of the setup we show that the Hilbert-Carleman determinant is a $\tau$-function of the Kadomtsev-Petviashvili (KP) or nonlinear Schr\"odinger hierarchies.