Schlesinger transformations for Painlevé VI equation

Muğan, Uğurhan; Sakka, A.

doi:10.1063/1.531121

Cited by 25 publications

(35 citation statements)

References 17 publications

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“…Moreover, it is easily verified that X α exhibits the following symmetry, 28) which yields the relation…”

Section: Recursive Construction Of ψmentioning

confidence: 99%

“…The crucial observation here is that the jump matrices for Ψ − α are periodic in α: they remain the same if we replace α by α + 2. This fact indicates that there is a Schlesinger type transformation [28] relating Ψ α to Ψ α+2 . Even more, if we replace α by α + 1, the jump matrices are modified in a simple way: the jumps on the diagonals Γ…”

Section: Recursive Construction Of ψmentioning

confidence: 99%

“…The limiting kernel K PV α (u, v; τ ) degenerates to the sine and Bessel kernels for large and small values of τ . Indeed, we will show that 28) and that…”

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confidence: 99%

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Random Matrices with Merging Singularities and the Painlevé V Equation

Claeys

Fahs²

2016

SIGMA

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Abstract. We study the asymptotic behavior of the partition function and the correlation kernel in random matrix ensembles of the formwhere M is an n × n Hermitian matrix, α > −1/2 and t ∈ R, in double scaling limits where n → ∞ and simultaneously t → 0. If t is proportional to 1/n 2 , a transition takes place which can be described in terms of a family of solutions to the Painlevé V equation. These Painlevé solutions are in general transcendental functions, but for certain values of α, they are algebraic, which leads to explicit asymptotics of the partition function and the correlation kernel.

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“…Moreover, it is easily verified that X α exhibits the following symmetry, 28) which yields the relation…”

Section: Recursive Construction Of ψmentioning

confidence: 99%

Section: Recursive Construction Of ψmentioning

confidence: 99%

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Random Matrices with Merging Singularities and the Painlevé V Equation

Claeys

Fahs²

2016

SIGMA

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“…Aux TS de P6 déjà connues [2,12,10] viennent de s'en ajouter deux nouvelles [11,1], et le but de cette Note est de déterminer la plusélémentaire de toutes ces TS.…”

Section: Introductionunclassified

Sur les transformations de Schlesinger de la sixième équation de Painlevé

Conte

2001

Comptes Rendus de l'Académie des Sciences - Series I - Mathemat

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Résumé. Après la récente découverte de deux nouvelles transformations de Schlesinger (TS) pour la sixièmeéquation de Painlevé, nous donnons les relations d'interdépendance de toutes les TS connues. Nous isolons ainsi l'unique d'entre elles quià la fois conserve la variable indépendante et n'est pas un produit d'autres TS.On the Schlesinger transformations of the sixth Painlevé equation Abstract. Following the recent discovery of two new Schlesinger transformations (ST) for the sixth Painlevé equation, we give the interrelations between all the known STs. We thus isolate the unique one which at the same time conserves the independent variable and is not a product of other STs.

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“…The general theory for these transformations in the framework of the isomonodromy deformations can be found in [20] and particular formulae for P 5 in [31].…”

Section: Alternate Fuchs-garnier Pairs For the Third Painlevé Equationmentioning

confidence: 99%

On the linearization of the Painlevé III–VI equations and reductions of the three-wave resonant system

Joshi

Kitaev

Treharne

2007

Journal of Mathematical Physics

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We extend similarity reductions of the coupled (2+1)-dimensional three-wave resonant interaction system to its Lax pair. Thus we obtain new 3 × 3 matrix Fuchs-Garnier pairs for the third, fourth, and fifth Painlevé equations, together with the previously known Fuchs-Garnier pair for the sixth Painlevé equation. These Fuchs-Garnier pairs have an important feature: they are linear with respect to the spectral parameter. Therefore we can apply the Laplace transform to study these pairs. In this way we found reductions of all pairs to the standard 2 × 2 matrix Fuchs-Garnier pairs obtained by M. Jimbo and T. Miwa. As an application of the 3 × 3 matrix pairs, we found an integral auto-transformation for the standard Fuchs-Garnier pair for the fifth Painlevé equation. It generates an Okamoto-like Bäcklund transformation for the fifth Painlevé equation. Another application is an integral transformation relating two different 2 × 2 matrix Fuchs-Garnier pairs for the third Painlevé equation.

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Schlesinger transformations for Painlevé VI equation

Abstract: A method to obtain the Schlesinger transformations for Painlevi VI equation is given. The procedure involves formulating a Riemann-Hilbert problem for a transformation matrix which transforms the solution of the linear problem but leaves the associated monodromy data the same.

Cited by 25 publications

References 17 publications

Random Matrices with Merging Singularities and the Painlevé V Equation

Random Matrices with Merging Singularities and the Painlevé V Equation

Sur les transformations de Schlesinger de la sixième équation de Painlevé

On the linearization of the Painlevé III–VI equations and reductions of the three-wave resonant system

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