2023/07/24 by Marco Bertola, Bertola, Marco, Тамара Грава +3
Mathematics · Physics and Astronomy · #Classical Analysis and ODEs (math.CA) #Differential Equations and Boundary Problems #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2307.13119
openalex publication_date 2023/07/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We develop the theory of integrable operators 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 ∂-problem in the complex plane. When such a ∂-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 K. With suitable choices of the setup we show that the Hilbert-Carleman determinant is a τ-function of the Kadomtsev-Petviashvili (KP) or nonlinear Schrödinger hierarchies.