Universal characters, integrable chains and the Painlevé equations

Tsuda, Teruhisa

doi:10.1016/j.aim.2004.10.016

Cited by 49 publications

(46 citation statements)

References 22 publications

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“…Typical examples of this class are obtained by applying Bäcklund transformations to the simple solutions characterized by the invariance with respect to the Dynkin diagram automorphisms. Many of such solutions are interpreted as simple specialization of the Schur functions or the universal characters [47,59,60,63,67,72,88,85,124].…”

Section: (1) Applying Bäcklund Transformations (Birational Transformamentioning

confidence: 99%

“…3 ) can be carried out by the simple substitution 124) and the limit ε → 0 which is denoted by i/ j −→ as shown in (8.120) (see also Fig.20). Remark 8.4.…”

Section: Degeneration Of Point Configurationsmentioning

confidence: 99%

“…Even in the second order equations, there are various discrete Painlevé equations which are not directly investigated in this paper, e.g., equations arising from the translations with different directions or length in the root lattice [55,109,121]. Also, we do not deal with higher order or multi-variable generalizations, which are now actively studied in relation with soliton equations [19,20,57,58,87,119,124,126], geometry of space of initial values [71,114], geometry of flag varieties [89], or general theory of monodromy preserving deformations [65]. The Painlevé equations are believed to define new transcendental functions, and it was rigorously proved for the Painlevé differential equations (see, for example [128,129]).…”

Section: Introductionmentioning

confidence: 99%

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Geometric aspects of Painlevé equations

Kajiwara¹,

Noumi²,

Yamada³

2017

J. Phys. A: Math. Theor.

124

250

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In this paper a comprehensive review is given on the current status of achievements in the geometric aspects of the Painlevé equations, with a particular emphasis on the discrete Painlevé equations. The theory is controlled by the geometry of certain rational surfaces called the spaces of initial values, which are characterized by eight point configuration on P 1 × P 1 and classified according to the degeneration of points. We give a systematic description of the equations and their various properties, such as affine Weyl group symmetries, hypergeometric solutions and Lax pairs under this framework, by using the language of Picard lattice and root systems. We also provide with a collection of basic data; equations, point configurations/root data, Weyl group representations, Lax pairs, and hypergeometric solutions of all possible cases.

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Section: (1) Applying Bäcklund Transformations (Birational Transformamentioning

confidence: 99%

“…3 ) can be carried out by the simple substitution 124) and the limit ε → 0 which is denoted by i/ j −→ as shown in (8.120) (see also Fig.20). Remark 8.4.…”

Section: Degeneration Of Point Configurationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Geometric aspects of Painlevé equations

Kajiwara¹,

Noumi²,

Yamada³

2017

J. Phys. A: Math. Theor.

124

250

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show abstract

“…Hence we can immediately obtain from Theorem 4.1 a class of algebraic solutions of q-P VI in terms of the universal character; cf. [24,26].…”

Section: The Solutions Of the Sixth Q-painlevé Equationmentioning

confidence: 99%

“…We refer to the result [26] where the higher order q-Painlevé equations also turn out to be certain similarity reductions of the q-UC hierarchy; cf. [24]. It is still an interesting open question why the universal character solves the Garnier system; see [25].…”

Section: Introductionmentioning

confidence: 99%

q-Painlevé VI Equation Arising from q-UC Hierarchy

Tsuda

Masuda

2005

Commun. Math. Phys.

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We study the q-difference analogue of the sixth Painlevé equation (q-P VI ) by means of tau functions associated with affine Weyl group of type D 5 . We prove that a solution of q-P VI coincides with a self-similar solution of the q-UC hierarchy. As a consequence, we obtain in particular algebraic solutions of q-P VI in terms of the universal character which is a generalization of Schur polynomial attached to a pair of partitions.

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Affine Weyl group symmetries of Frobenius Painlevé equations

Wang

2019

Math Methods in App Sciences

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In this paper, we introduce a Frobenius Painlevé IV equation and the corresponding Hamilton system, and we give the symmetric form of the Frobenius Painlevé IV equation. Then, we construct the Lax pair of the Frobenius Painlevé IV equation. Furthermore, we recall the Frobenius modified KP hierarchy and the Frobenius KP hierarchy by bilinear equations, then we show how to get Frobenius Painlevé IV equation from the Frobenius modified KP hierarchy. In order to study the different aspects of the Frobenius Painlevé IV equation, we give the similarity reduction and affine Weyl group symmetry of the equation. Similarly, we introduce a Frobenius Painlevé II equation and show the connection between the Frobenius modified KP hierarchy and the Frobenius Painlevé II equation.

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Universal characters, integrable chains and the Painlevé equations

Cited by 49 publications

References 22 publications

Geometric aspects of Painlevé equations

Geometric aspects of Painlevé equations

q-Painlevé VI Equation Arising from q-UC Hierarchy

Affine Weyl group symmetries of Frobenius Painlevé equations

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