Abstract
We consider a Pfaffian system expressing isomonodromy of an irregular system of Okubo type, depending on complex deformation parameters u=(u_1,ldots ,u_n), which are eigenvalues of the leading matrix at the irregular singularity. 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,ldots ,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 Balser et al. (I SIAM J Math Anal 12(5): 691–721, 1981) and Guzzetti (Funkcial Ekvac 59(3): 383–433, 2016) 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 (Balser et al. in I SIAM J Math Anal 12(5): 691–721, 1981; Guzzetti in Funkcial Ekvac 59(3): 383–433, 2016) to the isomonodromic case (with coalescences/confluences), allow to prove by means of Laplace transform the main result of Cotti et al. (Duke Math J arXiv:1706.04808, 2017), namely the analytic theory of non-generic isomonodromic deformations of the irregular system with coalescing eigenvalues.
Highlights
(III) The constant essential monodromy data can be computed from the system “frozen” at a fixed coalescence point
Theorem 5.1, which characterizes selected vector solutions and singular vector solutions of (1.4) and (5.3), so extending the results of [4] and [23] to the case depending on isomonodromic deformation parameters, including coalescing Fuchsian singularities u1, . . . , un
We study isomonodromy deformations of (1.4) when u varies in a polydisc containing a coalescence locus
Summary
Remark 1.1 If A(u) is holomorphic on the polydisc and (1.1) is an isomonodromic family on the polydisc minus the coalescence locus (in the sense of integrability of an associated Pfaffian system (2.14) introduced later), (1.2) are automatically satisfied and Theorem 1.1 of [13] holds. This is not mentioned in [13].
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