On the Painlevé Property of Isomonodromic Deformations of Fuchsian Systems

Побережный, Владимир Андреевич

doi:10.1007/s10440-008-9189-3

Cited by 5 publications

(3 citation statements)

References 8 publications

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“…That is, we consider the member of isomonodromy family of the BHC (1.3) at the exceptional point z = 0, y = 0, z/y = λ. Since all the isomonodromy integral curves in the yz−plane pass through the exceptional line of the blow up (z, z/y) → (z, y) (see [39]), no generality is lost.…”

Section:  mentioning

confidence: 99%

Invariant subspaces of biconfluent Heun operators and special solutions of Painlevé IV

Chiang¹,

Law²,

Yu³

2019

Preprint

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We show that there is a full correspondence between the parameters space of the degenerate biconfluent Heun connection (BHC) and that of Painlevé IV that admits special solutions. The BHC degenerates when either the Stokes' data for the irregular singularity at ∞ degenerates or the regular singular point at the origin becomes an apparent singularity. We show that if the BHC is written as isomonodromy family of biconfluent Heun equations (BHE), then the BHE degenerates precisely when it admits eigen-solutions of the biconfluent Heun operators, after choosing appropriate accessory parameter, of specially constructed invariant subspaces of finite dimensional solution spaces spanned by parabolic cylinder functions. We have found all eigensolutions over this parameter space apart from three exceptional cases after choosing the right accessory parameters. These eigen-solutions are expressed as certain finite sum of parabolic cylinder functions. We extend the above sum to new convergent series expansion in terms of parabolic cylinder functions to the BHE. The infinite sum solutions of the BHE terminates precisely when the parameters of the BHE assumes the same values as those of the degenerate biconfluent Heun connection except at three instances after choosing the right accessory parameter.

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Section:  mentioning

confidence: 99%

Invariant subspaces of biconfluent Heun operators and special solutions of Painlevé IV

Chiang¹,

Law²,

Yu³

2019

Preprint

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“…The system of PDEs dY = ωY is over-determined in general and does not have any local solutions. However, the flatness condition dω = ω ∧ ω guarantees the existence of local solutions of dY = ωY (see [24]). If one chooses a convenient basis so that A 3 is diagonal, such a flatness condition implies that X(t), the zero locus of the (1,2) entry of ω, satisfies the usual Painlevé VI equation [16, §3], [30, §2].…”

Section: The Correspondence To the Elliptic Painlevé VI Equationmentioning

confidence: 99%

Solutions of Darboux Equations, its Degeneration and Painlevé VI Equations

Chiang,

Ching,

Tsang

2019

Preprint

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In this paper, we study the Darboux equations in both classical and system form, which give the elliptic Painlevé VI equations by the isomonodromy deformation method. Then we establish the full correspondence between the special Darboux equations and the special Painlevé VI equations. Instead of the system form, we especially focus on the Darboux equation in a scalar form, which is the generalization of the classical Lamé equation. We introduce a new infinite series expansion (in terms of the compositions of hypergeometric functions and Jacobi elliptic functions) for the solutions of the Darboux equations and regard special solutions of the Darboux equations as those terminating series. The Darboux equations characterized in this manner have an almost (but not completely) full correspondence to the special types of the Painlevé VI equations. Finally, we discuss the convergence of these infinite series expansions. Contents 1 Introduction 1 2 The Darboux Connection and the Elliptic Painlevé VI Equation 4 2.

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“…Тем не менее можно показать, что пространство M * a естественным образом расслаивается на листы, для каждого из которых отображение RH a корректно определено (хотя и не на всем пространстве M a ; подробнее см. [60]). …”

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Различные Варианты Проблемы Римана - Гильберта Для Линейных Дифференциальных Уравнений

Gontsov¹,

Побережный²

2008

Uspekhi Mat. Nauk

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On the Painlevé Property of Isomonodromic Deformations of Fuchsian Systems

Cited by 5 publications

References 8 publications

Invariant subspaces of biconfluent Heun operators and special solutions of Painlevé IV

Invariant subspaces of biconfluent Heun operators and special solutions of Painlevé IV

Solutions of Darboux Equations, its Degeneration and Painlevé VI Equations

Различные Варианты Проблемы Римана - Гильберта Для Линейных Дифференциальных Уравнений

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