scholarly journals Riccati solutions of discrete Painlevé equations with Weyl group symmetry of type E8(1)

2003 ◽  
Vol 44 (3) ◽  
pp. 1396-1414 ◽  
Author(s):  
Mikio Murata ◽  
Hidetaka Sakai ◽  
Jin Yoneda
Keyword(s):  
Weyl Group ◽  
Painlevé Equations ◽  
Group Symmetry ◽  
Painleve Equations ◽  
Discrete Painlevé Equations

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 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 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.

Two variations on ( A 3 × A 1 × A 1 ) (1) type discrete Painlevé equations

2019 ◽  
Vol 475 (2229) ◽  
pp. 20190299
Author(s):  
Yang Shi
Keyword(s):  
Coxeter Groups ◽  
Infinite Order ◽  
Cluster Algebra ◽  
Painlevé Equations ◽  
Group Symmetry ◽  
Parabolic Subgroups ◽  
Reflection Groups ◽  
Kp Hierarchy ◽  
Painleve Equations ◽  
Discrete Painlevé Equations

By considering the normalizers of reflection subgroups of types A (1) 1 and A (1) 3 in W ~ ( D 5 ( 1 ) ) , two subgroups: W ~ ( A 3 × A 1 ) ( 1 ) ⋉ W ( A 1 ( 1 ) ) and W ~ ( A 1 × A 1 ) ( 1 ) ⋉ W ( A 3 ( 1 ) ) can be constructed from a ( A 3  ×  A 1  ×  A 1 ) (1) type subroot system. These two symmetries arose in the studies of discrete Painlevé equations (Kajiwara K, Noumi M, Yamada Y. 2002 q -Painlevé systems arising from q -KP hierarchy. Lett. Math. Phys. 62 , 259–268; Takenawa T. 2003 Weyl group symmetry of type D (1) 5 in the q -Painlevé V equation. Funkcial. Ekvac. 46 , 173–186; Okubo N, Suzuki T. 2018 Generalized q -Painlevé VI systems of type ( A 2 n +1  +  A 1  +  A 1 ) (1) arising from cluster algebra. ( http://arxiv.org/abs/quant-ph/1810.03252 )), where certain non-translational elements of infinite order were shown to give rise to discrete Painlevé equations. We clarify the nature of these elements in terms of Brink-Howlett theory of normalizers of Coxeter groups (Howlett RB. 1980 Normalizers of parabolic subgroups of reflection groups. J. London Math. Soc. (2) 21 , 62–80; Brink B, Howlett RB. 1999 Normalizers of parabolic subgroups in Coxeter groups. Invent. Math. 136 , 323–351). This is the first of a series of studies which investigates the properties of discrete integrable equations via the theory of normalizers.

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