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.

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 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.

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