Isomonodromic Laplace transform with coalescing eigenvalues and confluence of Fuchsian singularities

This talk is based to the paper [4], which is submitted to a journal and will be put on arXiv soon. 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 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 $u_1,\ldots,u_n$. 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. This extends a result of [1] and [3] 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 [1,3] to the isomonodromic case (with coalescences/confluences), allow to prove by means of Laplace transform the main result of [2], namely the analytic theory of non-generic isomonodromic deformations of the irregular system with coalescing eigenvalues.


[1] Balser W., Jurkat W.B., Lutz D.A., On the reduction of connection problems for differential equations with irregular singular points to ones with only regular singularities, I, SIAM J. Math. Anal. 12:5 (1981), 691–721.
[2] Cotti G., Dubrovin B., Guzzetti D., Isomonodromy deformations at an irregular singularity with coalescing eigenvalues, Duke Math. J. 168:6 (2019), 967–1108.
[3] Guzzetti D., On Stokes matrices in terms of connection coefficients, Funkcial. Ekvac. 59:3 (2016), 383–433.
[4] Guzzetti D., Isomonodromic Laplace transform with coalescing eigenvalues and confluence of Fuchsian singularities, submitted (2020).