Isomonodromic Laplace Transform with Coalescing Eigenvalues and Confluence of Fuchsian Singularities

TL;DR

This paper develops an isomonodromic Laplace transform framework for irregular systems with coalescing eigenvalues, extending existing theories to include confluence of singularities and providing new solutions and monodromy data.

Contribution

It introduces an isomonodromic Laplace transform for systems with coalescing eigenvalues, extending prior results to include confluence of singularities and constructing new solutions.

Findings

Constructed isomonodromic vector solutions and connection coefficients.

Extended the theory of isomonodromic deformations to include confluence.

Proved the main result on non-generic isomonodromic deformations using Laplace transform.

Abstract

We consider a Pfaffian system expressing isomonodromy of an irregular system of Okubo type, depending on complex deformation parameters u=(u_1,...,u_n), which are eigenvalues of the leading matrix at the irregular singuilarity. At the same time, we consider a Pfaffian system of non-normalized Schlesinger type expressing isomonodromy of a Fuchsian system, whose poles are the deformation parameters u_1,...,u_n. The parameters vary in a polydisc containing a coalescence locus for the eigenvalues of the leading matrix of the irregular system, corresponding to confluence of the Fuchsian singularities. We construct isomonodromic selected and singular vector solutions of the Fuchsian Pfaffian system together with their isomonodromic connection coefficients, so extending a result of references [4] and [20] to the isomonodromic case, including confluence of singularities. Then, we introduce an isomonodromic Laplace transform of the selected and singular vector solutions, allowing to obtain isomonodromic fundamental solutions for the irregular system, and their Stokes matrices expressed in terms of connection coefficients. These facts, in addition to extending [4] and [20] to the isomonodromic case with coalescences/confluences, allow to prove by means of Laplace transform the main result of reference [11], which is the analytic theory of non-generic isomonodromic deformations of the irregular system with coalescing eigenvalues.