Special Polynomials Associated with the Noumi-Yamada System of Type A5(1)

Masuda, Tetsu

doi:10.1619/fesi.48.231

Cited by 4 publications

(4 citation statements)

References 13 publications

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“…The structure of the roots of the special polynomials associated with rational solutions of P V , which is equivalent to the system of type A (1) 3 , is discussed by Clarkson [39] (see also [40]). The special polynomials associated with rational solutions of the system of type A (1) 5 is discussed by Masuda [49].…”

Section: Discussionmentioning

confidence: 99%

“…Umemura [22], see also [23,24], derived special polynomials with certain rational and algebraic solutions of P III , P V and P VI which have similar properties to the Yablonskii-Vorob'ev and Okamoto polynomials. Recently, there have been further studies of the special polynomials associated with rational solutions of P II [25][26][27][28][29][30][31][32], rational and algebraic solutions of P III [33][34][35][36], rational solutions of P IV [20,27,37,38], rational and algebraic solutions of P V [39][40][41], algebraic solutions of P VI [42][43][44][45][46], rational solutions of the Garnier system [47][48] and rational solutions of higher order Painlevé equations [49,50,52,53]; a comprehensive review is given in [54]. Several of these papers are concerned with the combinatorial structure and determinant representation of the polynomials, often related to the Hamiltonian structure and affine Weyl symmetries of the Painlevé equations.…”

Section: Introductionmentioning

confidence: 99%

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The Symmetric Fourth Painlevé Hierarchy and Associated Special Polynomials

Filipuk

Clarkson

2008

Stud Appl Math

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In this paper two families of rational solutions and associated special polynomials for the equations in the symmetric fourth Painlevé hierarchy are studied. The structure of the roots of these polynomials is shown to be highly regular in the complex plane. Further representations are given of the associated special polynomials in terms of Schur functions. The properties of these polynomials are compared and contrasted with the special polynomials associated with rational solutions of the fourth Painlevé equation.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The Symmetric Fourth Painlevé Hierarchy and Associated Special Polynomials

Filipuk

Clarkson

2008

Stud Appl Math

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“…that follow from (2) by inserting C = −1/2 and i = 0, and which can be obtained from the Hamiltonian (see e.g. [16,17,21])…”

Section: Review Of P Iii-v Equations and Their Symmetriesmentioning

confidence: 99%

Solutions of mixed Painlevé P_III—Vmodel

Alves

Aratyn

Gomes

et al. 2019

J. Phys. A: Math. Theor.

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We review the construction of the mixed Painlevé P III−V system in terms of a 4-boson integrable model and discuss its symmetries. Such a mixed system consist of an hybrid differential equation that for special limits of its parameters reduces to either Painlevé P III or P V . The aim of this paper is to describe solutions of P III−V model. In particular, we determine and classify rational, power series and transcendental solutions of P III−V . A class of power series solutions is shown to be convergent in accordance with the Briot-Bouquet theorem. Moreover, the P III−V equations are reduced to Riccati equations and solved for special values of parameters. The corresponding Riccati solutions can be expressed as Whittaker functions or alternatively confluent hypergeometric and Laguerre functions and are given by ratios of polynomials of order n when the parameter of P III−V equation is quantized by integer n ∈ Z.Painlevé P II equation. Painlevé P V equations appeared in the context of impenetrable Bose gas model [13]. More recently the scattering on two Aharonov-Bohm vortices was exactly solved in terms of solutions of the Painlevé III equation [7]. Ablowitz Ramani and Segur (ARS) showed a connection between the mKdV integrable model and the Painlevé P II equation when self-similarity limit was implemented. They suggested that such connection could be generalized to other integrable models [1] meaning that the self-similarity limit would lead for integrable models to equations with the Painlevè property. More recently, it was found [3] that the 2n boson integrable model obtained as particular reductions (Drinfeld-Sokolov) of KP integrable models connected to Toda lattice hierarchy gives rise to higher Painlevé equations invariant under extended affine Weyl groups. In particular, investigation of various Dirac reduction schemes applied to the 4-boson (n = 2) integrable model with Weyl symmetry structure A (1) 4 led to emergence of a new mixed P III−V model [4]. The second order P III−V equation for a canonical variable q is given by:

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“…The expression of rational solutions for g = 1, 2 in Theorem 2.6 has been established in a different manner by Masuda et al [11,12]. We refer also to the results [5,6,10,14,20,22,23,25,28] where a class of rational (or algebraic) solutions is expressed in terms of Schur polynomials, or universal characters, for each of other Painlevé equations and Garnier's generalizations.…”

Section: Rational Solutions Of P (A 2g+1 ) In Terms Of Universal Charmentioning

confidence: 99%

Universal characters, integrable chains and the Painlevé equations

Tsuda

2005

Advances in Mathematics

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The universal character is a generalization of the Schur polynomial attached to a pair of partitions; see (Adv. Math. 74 (1989) 57). We prove that the universal character solves the Darboux chain. The N-periodic closing of the chain is equivalent to the Painlevé equation of type A (1) N−1 . Consequently we obtain an expression of rational solutions of the Painlevé equations in terms of the universal characters.

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Special Polynomials Associated with the Noumi-Yamada System of Type A5(1)

Cited by 4 publications

References 13 publications

The Symmetric Fourth Painlevé Hierarchy and Associated Special Polynomials

The Symmetric Fourth Painlevé Hierarchy and Associated Special Polynomials

Solutions of mixed Painlevé P_III—Vmodel

Universal characters, integrable chains and the Painlevé equations

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Special Polynomials Associated with the Noumi-Yamada System of Type A5(1)

Cited by 4 publications

References 13 publications

The Symmetric Fourth Painlevé Hierarchy and Associated Special Polynomials

The Symmetric Fourth Painlevé Hierarchy and Associated Special Polynomials

Solutions of mixed Painlevé PIII—Vmodel

Universal characters, integrable chains and the Painlevé equations

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Product

Resources

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Solutions of mixed Painlevé P_III—Vmodel