R. Fuchs’ problem of the Painlevé equations from the first to the fifth

Ohyama, Yousuke; Okumura, Shoji

doi:10.1090/conm/593/11876

Cited by 7 publications

(13 citation statements)

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“…R. Fuchs' problem, see [OO2,St1], also concerns algebraic solutions of Painlevé equations. The second order linear differential equations resulting from this problem seem to be unrelated to the first variational equations.…”

Section: Introductionmentioning

confidence: 99%

“…The normal variational equation(s) of H along a given explicit solution are shown (Proposition 3.1) to be equivalent to the variational equation(s) for x ′′ = R(x ′ , x, t) along a given solution. For the equations P 2 to P 5 , there is a convenient list in [OO2] of all cases with algebraic solutions, up to Bäcklund transformations, for which variational equations can be examined. One expects, in accordance with [MR, MRS], that the variational equations produce differential Galois groups G such that G o , the component of the identity, is not abelian.…”

Section: Introductionmentioning

confidence: 99%

“…Clarkson's papers [C] and [C3, §5.6] contain a list of rational solutions of P 5 , e.g., P 5 ( 1 2 , − s 2 2 , 2 − s, − 1 2 ) with y = t + s. We skip these examples since they are equivalent via Bäcklund transformations to one of the two equations in [OO2,Theorem 2.1].…”

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confidence: 99%

“…In [OO2] another degenerate fifth Painlevé equation P 5 ( h 2 2 , −8, −2, 0) and solution y = 1 + 2t 1/2 h is considered. We translate this for the degenerate fifth Painlevé equation degP 5 of our paper [APT].…”

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Variations for Some Painlevé Equations

Acosta-Humánez¹,

Put

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2019

SIGMA

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This paper first discusses irreducibility of a Painleve equation P . We show that reducibility is equivalent to the existence of special classical solutions. As in a paper of Morales-Ruiz we associate an autonomous Hamiltonian H to a Painlevé equation P . Complete integrability of H is shown to imply that all solutions to P are special classical or algebraic, so in particular P is reducible.Next, we show that the variational equation of P at a given algebraic solution coincides with the normal variational equation of H at the corresponding solution.Finally, we test the Morales-Ramis theorem in all cases P 2 to P 5 where algebraic solutions are present, by showing how our results lead to a quick computation of the component of the identity of the differential Galois group for the first two variational equations. As expected there are no cases where this group is commutative.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

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Variations for Some Painlevé Equations

Acosta-Humánez¹,

Put

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2019

SIGMA

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“…Classical' papers on the subject are [8,9,12,13,14,15,16]. Especially relevant for the present text are the paper by Ohyama and Okumura [11], Witte's paper [26]. The book [2] by Fokas, Its, Kapaev, and Novokshenov discusses more analytic aspects of the Riemann-Hilbert correspondence, but it does not discuss the degenerate fifth Painlevé equation.…”

Section: Introductionmentioning

confidence: 99%

Isomonodromy for the Degenerate Fifth Painlevé Equation

Acosta-Humánez¹,

Put²,

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2017

SIGMA

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Abstract. This is a sequel to papers by the last two authors making the Riemann-Hilbert correspondence and isomonodromy explicit. For the degenerate fifth Painlevé equation, the moduli spaces for connections and for monodromy are explicitly computed. It is proven that the extended Riemann-Hilbert morphism is an isomorphism. As a consequence these equations have the Painlevé property and the Okamoto-Painlevé space is identified with a moduli space of connections. Using MAPLE computations, one obtains formulas for the degenerate fifth Painlevé equation, for the Bäcklund transformations.

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Computation of RS-Pullback Transformations for Algebraic Painlevé VI Solutions

Vidūnas

Kitaev

2016

J Math Sci

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Various Schlesinger transformations can be combined with a direct pull-back of a hypergeometric 2×2 system to obtain RS 2 4 -pullback transformations to isomonodromic 2 × 2 Fuchsian systems with 4 singularities. The corresponding Painlevé VI solutions are algebraic functions, possibly in different orbits under Okamoto transformations. This paper demonstrates a direct computation of Schlesinger transformations acting on several apparent singular points, and presents an algebraic procedure (via syzygies) of computing algebraic Painlevé VI solutions without deriving full RS-pullback transformations.An important observation is that the same rational covering R(x) can be used in several RS-pullback transformations. Application of different RS-transformations to respectively different matrix differential equations gives different algebraic Painlevé VI solutions. For example, [26] demonstrates usage of the same degree 10 covering to pullback three different hypergeometric equations (with the local monodromy differences 1/2, 1/3, k/7 with k = 1, 2 or 3) and obtain three algebraic Painlevé VI solutions unrelated by fractional-linear or Okamoto transformations.In our other concrete examples, we start with matrix hypergeometric equations with the icosahedral monodromy group. The pullbacked Fuchsian equations have the icosahedral monodromy group as well. Corresponding Painlevé VI solutions are called icosahedral [3]. There are 52 types of icosahedral Painlevé VI solutions in total [3], up to branching representation of the icosahedral monodromy group (or equivalently, up to Okamoto transformations). We recompute icosahedral solutions of Boalch types 26, 27, 31, 32. Second order ordinary Fuchsian equations (or 2 × 2 first order matrix Fuchsian equations) with a finite monodromy group are always pullbacks of a standard hypergeometric equation with the same monodromy group, as asserted by celebrated Klein's theorem [19]. In particular, existence of pull-back transformations for the four icosahedral examples follows from Klein's theorem. R. Fuchs [12] soon considered extension of Klein's theorem to algebraic solutions of Painlevé equations. Recently, Ohyama and Okumura [22] showed that algebraic solutions of Painlevé equations from the first to the fifth do arise from pullback transformations of confluent hypergeometric equations, affirming the formulation of R. Fuchs. The pullback method for computing algebraic Painlevé VI solutions was previously suggested in [18], [2], [17], [7]. The alternative representation-theoretic approach is due to Dubrovin-Mazzocco [9]. Recently, it was used [20] to complete classification of algebraic Painlevé VI solutions.The article is organized as follows. Section 2 presents two almost Belyi coverings we employ. The coverings have degree 8 and 12; they were previously used in [18]. Section 3 demonstrates two examples of full RS-pullback transformations, both with respect to the degree 8 covering. In Section 4 we formulate basic algebraic facts useful in computations of RS-pullback transformations....

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R. Fuchs’ problem of the Painlevé equations from the first to the fifth

Cited by 7 publications

References 20 publications

Variations for Some Painlevé Equations

Variations for Some Painlevé Equations

Isomonodromy for the Degenerate Fifth Painlevé Equation

Computation of RS-Pullback Transformations for Algebraic Painlevé VI Solutions

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