Schlesinger transformations for elliptic isomonodromic deformations

Korotkin, Dmitry; Manojlović, N.; Samtleben, Henning

doi:10.1063/1.533296

Cited by 16 publications

(22 citation statements)

References 21 publications

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“…For these purposes we develop a geometric techniques of the modifications ( [1]) of vector bundles with connections in the second and the third sections. In the original work [20] L. Schlesinger consider these transformations of the systems of isomonodromic deformations; further the algebraic aspects of Schlesinger transformations are developed in the paper of M. Jimba and T. Miwa [15] (see also [16]) but without paying attention to the group structure. Besides, in classical works [20] and [15] they discuss the monodromy representation in GL(N ).…”

Section: Introductionmentioning

confidence: 99%

Discrete Symmetries of Systems of Isomonodromic Deformations of Second-Order Fuchsian Differential Equations

Oblezin

2004

Functional Analysis and Its Applications

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In the present work we investigate the group structure of the Schlesinger transformations for isomonodromic deformations of order two Fuchsian differential equations. We perform these transformations as isomorphisms between the moduli spaces of the logarithmic sl(2)-connections with fixed eigenvalues of the residues at singular points. We give a geometrical interpretation of the Schlesinger transformations and perform our calculations using the techniques of the modifications of bundles with connections. In order to illustrate the result we present classical examples of symmetries of the hypergeometric equation, the Heun equation and the sixth Painlevé equation.

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Section: Introductionmentioning

confidence: 99%

Discrete Symmetries of Systems of Isomonodromic Deformations of Second-Order Fuchsian Differential Equations

Oblezin

2004

Functional Analysis and Its Applications

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“…Gaudin system [12,14] from the same point of view. Separation of variables of the isospectral partner has been studied by Sklyanin and Takebe [28] (see also the paper of Hurtubise and Kjiri [29] for geometric aspects).…”

Section: Discussionmentioning

confidence: 99%

“…and Olshanetsky[11] developed a general framework in which the Schlesinger system and Korotkin and Samtleben's isomonodromic system are placed, along with generalizations to higher genus Riemann surfaces, in a unified way. Some more examples of matrix systems with different structures are also known[12,13,14,15]. Compared with Okamoto and Iwasaki's formulation, these "elliptic analogues of the Schlesinger system" are obtained on an entirely different ground, such as conformal field theories, vector bundles on a torus, KZ equations, and (classical or quantum) integrable systems.…”

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confidence: 99%

Spectral curve and Hamiltonian structure of isomonodromic SU(2) Calogero–Gaudin system

Takasaki

2003

Journal of Mathematical Physics

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This paper presents a new approach to the Hamiltonian structure of isomonodromic deformations of a matrix system of ODE's on a torus. An isomonodromic analogue of the SU(2) Calogero-Gaudin system is used for a case study of this approach. A clue of this approach is a mapping to a finite number of points on the spectral curve of the isomonodromic Lax equation. The coordinates of these moving points give a new set of Darboux coordinates called the spectral Darboux coordinates. The system of isomonodromic deformations is thereby converted to a non-autonomous Hamiltonian system in the spectral Darboux coordinates. TheHamiltonians turn out to resemble those of a previously known isomonodromic system of a second order scalar ODE. The two isomonodromic systems are shown to be linked by a simple relation.

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“…A first step in this direction was recently made in Refs. [31,32,33], who relaxed the prescription to shifts by half-integers, so as to globally change the monodromy matrix to its opposite rather than to conserve it.…”

Section: On Schlesinger Transformationsmentioning

confidence: 99%

New contiguity relation of the sixth Painlevé equation from a truncation

Conte

Musette

2002

Physica D: Nonlinear Phenomena

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For the master Painlevé equation P6(u), we define a consistent method, adapted from the Weiss truncation for partial differential equations, which allows us to obtain the first degree birational transformation of Okamoto. Two new features are implemented to achieve this result. The first one is the homography between the derivative of the solution u and a Riccati pseudopotential. The second one is an improvement of a conjecture by Fokas and Ablowitz on the structure of this birational transformation. We then build the contiguity relation of P6, which yields one new second order nonautonomous discrete equation.

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Schlesinger transformations for elliptic isomonodromic deformations

Cited by 16 publications

References 21 publications

Discrete Symmetries of Systems of Isomonodromic Deformations of Second-Order Fuchsian Differential Equations

Discrete Symmetries of Systems of Isomonodromic Deformations of Second-Order Fuchsian Differential Equations

Spectral curve and Hamiltonian structure of isomonodromic SU(2) Calogero–Gaudin system

New contiguity relation of the sixth Painlevé equation from a truncation

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