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 Remarks on the Yablonskii-Vorob’ev polynomials

2000 ◽  
Vol 159 ◽  
pp. 87-111 ◽  
Author(s):  
Makoto Taneda
Keyword(s):  
Algebraic Approach ◽  
Painlevé Equations ◽  
Painlevé Equation ◽  
Painleve Equations ◽  
Special Polynomials ◽  
Painleve Equation ◽  
Second Painlevé Equation

We study the Yablonskii-Vorob’ev polynomial associated with the second Painlevé equation. To study other special polynomials (Okamoto polynomials, Umemura polynomials) associated with the Painlevé equations, our purely algebraic approach is useful.

scholarly journals Exact solutions of a q -discrete second Painlevé equation from its iso-monodromy deformation problem. II. Hypergeometric solutions

2012 ◽  
Vol 468 (2146) ◽  
pp. 3247-3264 ◽  
Author(s):  
Nalini Joshi ◽  
Yang Shi
Keyword(s):  
Linear Problem ◽  
Infinite Sequence ◽  
Painlevé Equation ◽  
Rational Solutions ◽  
Determinantal Representation ◽  
Deformation Problem ◽  
Hypergeometric Type ◽  
Painleve Equation ◽  
Second Painlevé Equation ◽  
Hypergeometric Solutions

This is the second part of our study of the solutions of a q -discrete second Painlevé equation ( q -P II ) of type ( A 2 + A 1 ) (1) via its iso-monodromy deformation problem. In part I, we showed how to use the q -discrete linear problem associated with q -P II to find an infinite sequence of exact rational solutions. In this paper, we study the case giving rise to an infinite sequence of q -hypergeometric-type solutions. We find a new determinantal representation of all such solutions and solve the iso-monodromy deformation problem in closed form.

scholarly journals ON A NOVEL CLASS OF INTEGRABLE ODEs RELATED TO THE PAINLEVÉ EQUATIONS

2012 ◽  
Vol 22 (09) ◽  
pp. 1250211
Author(s):  
ATHANASSIOS S. FOKAS ◽  
DI YANG
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Lax Pair ◽  
Painlevé Equations ◽  
Painlevé Equation ◽  
Painleve Equations ◽  
Independent Variable ◽  
Painlevé Ii ◽  
Painleve Equation

One of the authors recently introduced the concept of conjugate Hamiltonian systems: the solution of the equation h = H(p, q, t), where H is a given Hamiltonian containing t explicitly, yields the function t = T(p, q, h), which defines a new Hamiltonian system with Hamiltonian T and independent variable h. By employing this construction and by using the fact that the classical Painlevé equations are Hamiltonian systems, it is straightforward to associate with each Painlevé equation two new integrable ODEs. Here, we investigate the conjugate Painlevé II equations. In particular, for these novel integrable ODEs, we present a Lax pair formulation, as well as a class of implicit solutions. We also construct conjugate equations associated with Painlevé I and Painlevé IV 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.

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.

scholarly journals Classical solutions of the third Painlevé equation

1995 ◽  
Vol 139 ◽  
pp. 37-65 ◽  
Author(s):  
Yoshihiro Murata
Keyword(s):  
Painlevé Equations ◽  
Painlevé Equation ◽  
Classical Solutions ◽  
Painleve Equations ◽  
The Third ◽  
Painleve Equation

The big problem “Do Painlevé equations define new functions?”, what is called the problem of irreducibilities of Painlevé equations, was essentially solved by H. Umemura [16], [17] and K. Nishioka [9].

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