Isomonodromic Laplace transform with coalescing eigenvalues and confluence of Fuchsian singularities

Abstract

We consider a Pfaffian system expressing isomonodromy of an irregular system of Okubo type, depending on complex deformation parameters $$u=(u_1,\ldots ,u_n)$$ u = ( u 1 , … , 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 $$u_1,\ldots ,u_n$$ 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 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.

The sixth Painleve' equation as isomonodromy deformation of an irregular system: monodromy data, coalescing eigenvalues, locally holomorphic transcendents and Frobenius manifolds

“…While in [35] both the irregular system (1.4) and the Fuchsian system have the same dimension, in our case the irregular system (1.4) has n " 3 , while the Fuchsian system (1.2) is 2-dimensional. Thus, in the proof of Theorem 3.1 we will exploit the results of [35] in order to properly express the monodromy matrices of the 2-dimensional system (1.2) in terms of the connection coefficients of the 3-dimensional Fuchsian system associated with (1.4) by Laplace transform. The main technical difficulty will be that the selected solutions of the 3-dimensional Fuchsian system may be not linearly independent.…”

“…The proof of Theorem 3.1 is based on the isomonodromic Laplace transform [35], relating an n dimensional isomonodromic system of the type (1.4) to an isomonodromic Fuchsian system of the same dimension, with poles at λ " u k , k " 1, ..., n. The latter admits certain selected vector solutions, whose monodromy can be written in terms of certain connection coefficients. In [35], the connection coefficients are expressed as linear functions of the entries of the Stokes matrices of the n-dimensional (1.4) and vice-versa, including the coalescent case. While in [35] both the irregular system (1.4) and the Fuchsian system have the same dimension, in our case the irregular system (1.4) has n " 3 , while the Fuchsian system (1.2) is 2-dimensional.…”

“…In [35], the connection coefficients are expressed as linear functions of the entries of the Stokes matrices of the n-dimensional (1.4) and vice-versa, including the coalescent case. While in [35] both the irregular system (1.4) and the Fuchsian system have the same dimension, in our case the irregular system (1.4) has n " 3 , while the Fuchsian system (1.2) is 2-dimensional. Thus, in the proof of Theorem 3.1 we will exploit the results of [35] in order to properly express the monodromy matrices of the 2-dimensional system (1.2) in terms of the connection coefficients of the 3-dimensional Fuchsian system associated with (1.4) by Laplace transform.…”

“…As a consequence of theorem 1.1. of [7], in order to compute the monodromy data of system (1.4), it suffices to compute the data of dY dz " ˆU pu c q `V pu c q z ˙Y, (1.12) which is simpler than (1.4) at generic u. 4 The above results have been re-obtained in [35] using the isomonodromic Laplace transform (this is the point of view used here to prove Theorem 3.1).…”

Isomonodromic deformations along a stratum of the coalescence locus

“…In the work [13], and in the related [12,18,19,20,21], we have studied an n ˆn matrix differential system of the shape (1.1) below, with an irregular singularity at z " 8 and a Fuchsian one at z " 0, whose leading term at 8 is a diagonal matrix Λ " diagpu 1 , ..., u n q, whose eignevalus u " pu 1 , ..., u n q vary in a polydisc of C n . The polydisc contains a coalescence locus, where some eigenvalues merge, namely u j ´uk Ñ 0 for some j ‰ k. For this system, we have proved that a monodromy preserving deformation theory can be well defined (in an analytic way) with constant monodromy data on the whole polydisc, including the coalescence locus.…”