scholarly journals Symmetries in the fourth Painlevé equation and Okamoto polynomials

1999 ◽  
Vol 153 ◽  
pp. 53-86 ◽  
Author(s):  
Masatoshi Noumi ◽  
Yasuhiko Yamada
Keyword(s):  
Weyl Group ◽  
Schur Functions ◽  
Painlevé Equation ◽  
Symmetric Form ◽  
Rational Solutions ◽  
Kp Hierarchy ◽  
Similarity Reduction ◽  
Affine Weyl Group ◽  
Dependent Variables ◽  
Painleve Equation

AbstractThe fourth Painlevé equation PIV is known to have symmetry of the affine Weyl group of type with respect to the Bäcklund transformations. We introduce a new representation of PIV, called the symmetric form, by taking the three fundamental invariant divisors as the dependent variables. A complete description of the symmetry of PIV is given in terms of this representation. Through the symmetric form, it turns out that PIV is obtained as a similarity reduction of the 3-reduced modified KP hierarchy. It is proved in particular that the special polynomials for rational solutions PIV, called Okamoto polynomials, are expressible in terms of the 3-reduced Schur functions.

scholarly journals On the transformation group of the second Painlevé equation

2000 ◽  
Vol 157 ◽  
pp. 15-46 ◽  
Author(s):  
Hiroshi Umemura
Keyword(s):  
Weyl Group ◽  
Transformation Group ◽  
Blow Up ◽  
Initial Conditions ◽  
Painlevé Equation ◽  
Affine Weyl Group ◽  
Painleve Equation ◽  
Second Painlevé Equation ◽  
Projective Lines ◽  
Projective Surfaces

We show that for the second Painlevé equation y″ = 2y3 + ty + α, the Bäcklund transformation group G, which is isomorphic to the extended affine Weyl group of type Â1, operates regularly on the natural projectification χ(c)/ℂ(c, t) of the space of initial conditions, where c = α - 1/2. χ(c)/ℂ(c, t) has a natural model χ[c]/ℂ(t)[c]. The group G does not operate, however, regularly on χ[c]/ℂ(t)[c]. To have a family of projective surfaces over ℂ(t)[c] on which G operates regularly, we have to blow up the model χ[c] along the projective lines corresponding to the Riccati type solutions.

QUANTUM PAINLEVÉ SYSTEMS OF TYPE $A_l^{(1)}$

2004 ◽  
Vol 15 (10) ◽  
pp. 1007-1031 ◽  
Author(s):  
HAJIME NAGOYA
Keyword(s):  
Weyl Group ◽  
Discrete Systems ◽  
Painlevé Equation ◽  
Differential Systems ◽  
Type Formula ◽  
Affine Weyl Group ◽  
Painleve Equation ◽  
Second Painlevé Equation ◽  
Group Symmetries

We propose quantum Painlevé systems of type [Formula: see text]. These systems, for l=1 and l≥2, should be regarded as quantizations of the second Painlevé equation and the differential systems with the affine Weyl group symmetries of type [Formula: see text] studied by Noumi and Yamada [13], respectively. These quantizations enjoy the affine Weyl group symmetries of type [Formula: see text] as well as the Lax representations. The quantized systems of type [Formula: see text] and type [Formula: see text](l=2n) can be obtained as the continuous limits of the discrete systems constructed from the affine Weyl group symmetries of type [Formula: see text] and [Formula: see text], respectively.

scholarly journals Exact solutions of a q -discrete second Painlevé equation from its iso-monodromy deformation problem: I. Rational solutions

2011 ◽  
Vol 467 (2136) ◽  
pp. 3443-3468 ◽  
Author(s):  
Nalini Joshi ◽  
Yang Shi
Keyword(s):  
Exact Solutions ◽  
Linear Problem ◽  
Painlevé Equations ◽  
Painlevé Equation ◽  
Group Symmetry ◽  
Rational Solutions ◽  
Painleve Equations ◽  
Discrete Painlevé Equations ◽  
Painleve Equation ◽  
Second Painlevé Equation

In this paper, we present a new method of deducing infinite sequences of exact solutions of q -discrete Painlevé equations by using their associated linear problems. The specific equation we consider in this paper is a q -discrete version of the second Painlevé equation ( q -P II ) with affine Weyl group symmetry of type ( A 2 + A 1 ) (1) . We show, for the first time, how to use the q -discrete linear problem associated with q -P II to find an infinite sequence of exact rational solutions and also show how to find their representation as determinants by using the linear problem. The method, while demonstrated for q -P II here, is also applicable to other discrete Painlevé equations.

scholarly journals The down operator and expansions of near rectangular k-Schur functions

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Chris Berg ◽  
Franco Saliola ◽  
Luis Serrano
Keyword(s):  
Weyl Group ◽  
Schur Functions ◽  
Nous Obtenons ◽  
Combinatorial Interpretation ◽  
Affine Weyl Group ◽  
Nous Montrons ◽  
International Audience

International audience We prove that the Lam-Shimozono ``down operator'' on the affine Weyl group induces a derivation of the affine Fomin-Stanley subalgebra of the affine nilCoxeter algebra. We use this to verify a conjecture of Berg, Bergeron, Pon and Zabrocki describing the expansion of k-Schur functions of ``near rectangles'' in the affine nilCoxeter algebra. Consequently, we obtain a combinatorial interpretation of the corresponding k-Littlewood–Richardson coefficients. Nous montrons que l’opérateur ``down'', défini par Lam et Shimozono sur le groupe de Weyl affine, induit une dérivation de la sous-algèbre affine de Fomin-Stanley de l'algèbre affine de nilCoxeter. Nous employons cette dérivation pour vérifier une conjecture de Berg, Bergeron, Pon et Zabrocki sur l'expansion des k-fonctions de Schur indexées par les partitions qui sont ``presque rectangles''. Par conséquent, nous obtenons une interprétation combinatoire des k-coefficients de Littlewood–Richardson correspondants.

Special polynomials associated with rational solutions of the defocusing nonlinear Schrödinger equation and the fourth Painlevé equation

2006 ◽  
Vol 17 (3) ◽  
pp. 293-322 ◽  
Author(s):  
PETER A. CLARKSON
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Painlevé Equation ◽  
Rational Solutions ◽  
Oscillatory Solutions ◽  
Nonlinear Schrödinger ◽  
Special Polynomials ◽  
Painleve Equation

Rational solutions and rational-oscillatory solutions of the defocusing nonlinear Schrödinger equation are expressed in terms of special polynomials associated with rational solutions of the fourth Painlevé equation. The roots of these special polynomials have a regular, symmetric structure in the complex plane. The rational solutions verify results of Nakamura and Hirota [J. Phys. Soc. Japan, 54 (1985) 491–499] whilst the rational-oscillatory solutions appear to be new solutions of the defocusing nonlinear Schrödinger equation.

scholarly journals Generalized q-Painlevé VI Systems of Type (A2n+1+A1+A1)(1) Arising From Cluster Algebra

2020 ◽  
Author(s):  
Naoto Okubo ◽  
Takao Suzuki
Keyword(s):  
Weyl Group ◽  
Cluster Algebra ◽  
Higher Order ◽  
Root Lattice ◽  
Similarity Reduction ◽  
Birational Transformations ◽  
System A ◽  
Affine Weyl Group ◽  
Painlevé Vi Equation ◽  
Garnier System

Abstract In this article we formulate a group of birational transformations that is isomorphic to an extended affine Weyl group of type $(A_{2n+1}+A_1+A_1)^{(1)}$ with the aid of mutations and permutations of vertices to a mutation-periodic quiver on a torus. This group provides a class of higher order generalizations of Jimbo–Sakai’s $q$-Painlevé VI equation as translations on a root lattice. Then the known three systems are obtained again: the $q$-Garnier system, a similarity reduction of the lattice $q$-UC hierarchy, and a similarity reduction of the $q$-Drinfeld–Sokolov hierarchy.

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