Affine Weyl group symmetries of Frobenius Painlevé equations

2019 ◽  
Vol 43 (6) ◽  
pp. 3238-3252 ◽  
Author(s):  
Haifeng Wang ◽  
Chuanzhong Li
Keyword(s):  
Weyl Group ◽  
Painlevé Equations ◽  
Painleve Equations ◽  
Affine Weyl Group ◽  
Group Symmetries

QUANTUM PAINLEVÉ SYSTEMS OF TYPE $A_{n-1}^{(1)}$ WITH HIGHER DEGREE LAX OPERATORS

2007 ◽  
Vol 18 (07) ◽  
pp. 839-868 ◽  
Author(s):  
HAJIME NAGOYA
Keyword(s):  
Weyl Group ◽  
Heisenberg Algebra ◽  
Painlevé Equations ◽  
Differential Systems ◽  
Classical Systems ◽  
Painleve Equations ◽  
Type Formula ◽  
Affine Weyl Group ◽  
Lax Operator ◽  
Group Symmetries

Quantum Painlevé systems of type [Formula: see text] [13] are the quantizations of the second, fourth and fifth Painlevé equations and their generalizations [1, 15, 26]. These quantized systems have the Lax representations as in the classical systems. As a polynomial in an element of a Heisenberg algebra of [Formula: see text], the degrees of those Lax operators are 2 or 3. In this paper, we shall deal with the Lax operator whose degree is greater than or equal to 2. Using this Lax operator, we systematically construct the differential systems with the affine Weyl group symmetries of type [Formula: see text] and the commuting Hamiltonians.

scholarly journals Lax pairs of discrete Painlevé equations: ( A 2 + A 1 ) (1) case

2016 ◽  
Vol 472 (2196) ◽  
pp. 20160696 ◽  
Author(s):  
Nalini Joshi ◽  
Nobutaka Nakazono
Keyword(s):  
Weyl Group ◽  
Painlevé Equations ◽  
Group Symmetry ◽  
Lax Pairs ◽  
Painleve Equations ◽  
Comprehensive Method ◽  
Affine Weyl Group ◽  
Discrete Painlevé Equations

In this paper, we provide a comprehensive method for constructing Lax pairs of discrete Painlevé equations by using a reduced hypercube structure. In particular, we consider the A 5 ( 1 ) -surface q -Painlevé system, which has the affine Weyl group symmetry of type ( A 2 + A 1 ) (1) . Two new Lax pairs are found.

scholarly journals FROM KP/UC HIERARCHIES TO PAINLEVÉ EQUATIONS

2012 ◽  
Vol 23 (05) ◽  
pp. 1250010 ◽  
Author(s):  
TERUHISA TSUDA
Keyword(s):  
Weyl Group ◽  
Integrable Systems ◽  
Painlevé Equations ◽  
Group Symmetry ◽  
Schur Function ◽  
Infinite Dimensional ◽  
Painleve Equations ◽  
Bilinear Relations ◽  
Algebraic Solutions ◽  
Hirota Bilinear

We study the underlying relationship between Painlevé equations and infinite-dimensional integrable systems, such as the KP and UC hierarchies. We show that a certain reduction of these hierarchies by requiring homogeneity and periodicity yields Painlevé equations, including their higher order generalization. This result allows us to clearly understand various aspects of the equations, e.g. Lax formalism, Hirota bilinear relations for τ-functions, Weyl group symmetry and algebraic solutions in terms of the character polynomials, i.e. the Schur function and the universal character.

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.

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