General linear problem of the isomonodromic deformation of Fuchsian systems

Побережный, Владимир Андреевич

doi:10.1134/s0001434607030303

Cited by 8 publications

(6 citation statements)

References 3 publications

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“…The proof can be found in ( [12]). Parameters t can be considered as the coordinates on the moduli space continuous component.…”

Section: Theoremmentioning

confidence: 96%

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On the Painlevé Property of Isomonodromic Deformations of Fuchsian Systems

Побережный¹

2008

Acta Appl Math

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We give a review of the modern theory of isomonodromic deformations of Fuchsian systems discussing both classical and modern results, such as a general form of the isomonodromic deformations of Fuchsian systems, their differences from the classical Schlesinger deformations, the Fuchsian system moduli space structure and the geometric meaning of new degrees of freedom appeared in a non-Schlesinger case. Using this we illustrate some general relations between such concepts as integrability, isomonodromy and Painlevé property.

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“…The proof can be found in ( [12]). Parameters t can be considered as the coordinates on the moduli space continuous component.…”

Section: Theoremmentioning

confidence: 96%

“…If it is still identical then it is a Schlesinger deformation. Otherwise there is some additional dynamics over the continuous component of moduli space of Fuchsian systems that is a non-Schlesinger deformation ( [12]). …”

Section: Theorem 4 (Rigidity) For Any Given Set Of Singular Points Mmentioning

confidence: 99%

On the Painlevé Property of Isomonodromic Deformations of Fuchsian Systems

Побережный¹

2008

Acta Appl Math

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“…The resulting Fuchsian system remains resonant. See also [42] for other special transformations of Fuchsian systems.…”

Section: Non-schlesinger Isomonodromic Deformations and Middle Convol...mentioning

confidence: 99%

“…There exist some further studies and extensions of the notion of isomonodromic deformations of the Fuchsian systems (see, for instance, [38,42]). In this paper we deal with the definition of isomonodromic deformations as in Definition 1, Theorems 1 and 2.…”

Section: Introductionmentioning

confidence: 99%

Non-Schlesinger Isomonodromic Deformations of Fuchsian Systems and Middle Convolution

Bibilo

Filipuk

2015

SIGMA

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Abstract. The paper is devoted to non-Schlesinger isomonodromic deformations for resonant Fuchsian systems. There are very few explicit examples of such deformations in the literature. In this paper we construct a new example of the non-Schlesinger isomonodromic deformation for a resonant Fuchsian system of order 5 by using middle convolution for a resonant Fuchsian system of order 2. Moreover, it is known that middle convolution is an operation that preserves Schlesinger's deformation equations for non-resonant Fuchsian systems. In this paper we show that Bolibruch's non-Schlesinger deformations of resonant Fuchsian systems are, in general, not preserved by middle convolution.

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“…(Further refinements of the concept of isomonodromic deformation of Fuchsian systems, developing the notions introduced by Bolibrukh, were obtained by Anosov [54].) Examples of non-Schlesinger deformations can be found in [55].…”

Section: Isomonodromic Deformations Of Meromorphic Systems Isomonodrmentioning

confidence: 99%

Andrei Andreevich Bolibrukh's works on the analytic theory of differential equations

Anosov¹,

Leksin²

2011

Russ. Math. Surv.

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This paper contains an account of A. A. Bolibrukh's results obtained in the new directions of research that arose in the analytic theory of differential equations as a consequence of his sensational counterexample to the Riemann-Hilbert problem. A survey of results of his students in developing topics first considered by Bolibrukh is also presented. The main focus is on the role of the reducibility/irreducibility of systems of linear differential equations and their monodromy representations. A brief synopsis of results on the multidimensional Riemann-Hilbert problem and on isomonodromic deformations of Fuchsian systems is presented, and the main methods in the modern analytic theory of differential equations are sketched.Bibliography: 69 titles.Keywords: regular and Fuchsian systems of linear differential equations, monodromy representations of meromorphic systems of differential equations, Riemann-Hilbert problem, reducible and irreducible monodromy representations and systems of differential equations, isomonodromic deformations.

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General linear problem of the isomonodromic deformation of Fuchsian systems

Cited by 8 publications

References 3 publications

On the Painlevé Property of Isomonodromic Deformations of Fuchsian Systems

On the Painlevé Property of Isomonodromic Deformations of Fuchsian Systems

Non-Schlesinger Isomonodromic Deformations of Fuchsian Systems and Middle Convolution

Andrei Andreevich Bolibrukh's works on the analytic theory of differential equations

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