Linear Differential Equations in the Complex Domain

Haraoka, Yoshishige

doi:10.1007/978-3-030-54663-2

Cited by 16 publications

(22 citation statements)

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“…as we wanted to prove. Finally, the fact that Ψ pp 1 q pλ, uq has analytic continuation on the universal covering of P η puq ˆDpu c q follows from general results in the theory of linear Pfaffian systems [23,27,44].…”

Section: Fundamental Matrix Solution Of the Pfaffian Systemmentioning

confidence: 99%

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Isomonodromic Laplace Transform with Coalescing Eigenvalues and Confluence of Fuchsian Singularities

Guzzetti

2021

Preprint

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We consider a Pfaffian system expressing isomonodromy of an irregular system of Okubo type, depending on complex deformation parameters u " pu1, ..., unq, 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 u1, ..., un. 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 [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 [4,20] to the isomonodromic case (with coalescences/confluences), allow to prove by means of Laplace transform the main result of [11], which is the analytic theory of non-generic isomonodromic deformations of the irregular system with coalescing eigenvalues.

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Section: Fundamental Matrix Solution Of the Pfaffian Systemmentioning

confidence: 99%

“…Thus, there exists a constant matrix G such that G ´1B i G is a constant Jordan form, and G piq puq :" GpuqG realises the holomorphic "Jordanization" . The above arguments are standard, see for example [23].…”

Section: 4mentioning

confidence: 99%

Isomonodromic Laplace Transform with Coalescing Eigenvalues and Confluence of Fuchsian Singularities

Guzzetti

2021

Preprint

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“…as we wanted to prove. Finally, the fact that ( p 1 ) (λ, u) has analytic continuation on the universal covering of P η (u)×D(u c ) follows from general results in the theory of linear Pfaffian systems [28,32,63].…”

Section: Proofmentioning

confidence: 99%

“…Thus, there exists a constant matrix G such that G −1 B i G is a constant Jordan form, and G (i) (u) := G(u)G realizes the holomorphic "Jordanization". The above arguments are standard, see for example [28].…”

Section: Non-uniqueness At U = U C Of the Formal Solutionmentioning

confidence: 99%

Isomonodromic Laplace transform with coalescing eigenvalues and confluence of Fuchsian singularities

Guzzetti

2021

Lett Math Phys

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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.

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“…• the analiticity of the solutions of a compatible Pfaffian systems with polynomial coefficients (see for instance [15]). • the invariance of the space of solutions with respect to scaling transformations u i → c d i u i .…”

Section: Summarizing We Havementioning

confidence: 99%

A Dubrovin-Frobenius manifold structure of NLS type on the orbit space of $B_n$

Arsie,

Lorenzoni,

Mencattini

et al. 2021

Preprint

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Generalizing a construction presented in [3], we show that the orbit space of B 2 less the image of coordinate lines under the quotient map is equipped with two Dubrovin-Frobenius manifold structures which are related respectively to the defocusing and the focusing nonlinear Schr ödinger (NLS) equations. Motivated by this example, we study the case of B n and we show that the defocusing case can be generalized to arbitrary n leading to a Dubrovin-Frobenius manifold structure on the orbit space of the group. The construction is based on the existence of a non-degenerate and non-constant invariant bilinear form that plays the role of the Euclidean metric in the Dubrovin-Saito standard setting. Up to n = 4 the prepotentials we get coincide with those associated with constrained KP equations discussed in [20]. CONTENTS 1. Introduction 2. Bi-flat F -manifolds and Frobenius manifolds 3. Bi-flat F-manifolds and complex reflection groups 3.1. Flat structures associated with Coxeter groups 3.2. Dubrovin-Saito construction for B 2 3.3. Almost dual structure and Veselov's ∨-system 3.4. Flat structures associated with complex reflection groups 3.5. A simple example: B 2 4. A modified construction 4.1. The case of B 2 4.2. The cases B 3 and B 4 5. A flat pencil of metrics associated with B n 5.1. Invariance of g with respect to the action of B n 5.2. Flatness of g 5.3. Definition of η 5.4. The pair (g, η) is a flat pencil of metrics 6. Dubrovin-Frobenius structure on C n /B n 6.1. From flat pencils of metrics to Dubrovin-Frobenius manifolds 6.2. Step 1: definition of the c i jk s

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Linear Differential Equations in the Complex Domain

Cited by 16 publications

References 0 publications

Isomonodromic Laplace Transform with Coalescing Eigenvalues and Confluence of Fuchsian Singularities

Isomonodromic Laplace Transform with Coalescing Eigenvalues and Confluence of Fuchsian Singularities

Isomonodromic Laplace transform with coalescing eigenvalues and confluence of Fuchsian singularities

A Dubrovin-Frobenius manifold structure of NLS type on the orbit space of $B_n$

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