Lax pairs of discrete Painlevé equations: (A2+A1)(1)case

Joshi, Nalini; Nakazono, Nobutaka

doi:10.1098/rspa.2016.0696

Cited by 9 publications

(5 citation statements)

References 60 publications

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“…First, Lax pairs for discrete Painlevé equations can be derived using Lax pairs of the quad-equations [13,14]. Second, higher dimensional generalisations of discrete Painlevé equations can be obtained by generalising the combinatorics of the associated quad-equations and the reduction conditions.…”

Section: Resultsmentioning

confidence: 99%

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Reflection groups and discrete integrable systems

2015

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Abstract. We present a method of constructing discrete integrable systems with crystallographic reflection group (Weyl) symmetries, thus clarifying the relationship between different discrete integrable systems in terms of their symmetry groups. Discrete integrable systems are associated with space-filling polytopes arise from the geometric representation of the Weyl groups in the n-dimensional real Euclidean space R n . The "multi-dimensional consistency" property of the discrete integrable system is shown to be inherited from the combinatorial properties of the polytope; while the dynamics of the system is described by the affine translations of the polytopes on the weight lattices of the Weyl groups. The connections between some well-known discrete systems such as the multi-dimensional consistent systems of quad-equations [1] and discrete Painlevé equations [28] are obtained via the geometric constraints that relate the polytope of one symmetry group to that of another symmetry group, a procedure which we call geometric reduction.

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Section: Resultsmentioning

confidence: 99%

“…For x 1 and x 3 use Equations (3.19b) and (3.19a), respectively. For x 13 , we use what is called the "tetrahedron equation" (see the tetrahedron outlined by the densely dotted lines in Figure 3.1.3 = 1 implies Equation (3.24).…”

Section: Proposition 35 ([1]) Given a Hmentioning

confidence: 99%

Reflection groups and discrete integrable systems

2015

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“…It was shown in [4] that these translations act as Schlesinger transformations on the spectral equation (1.3a). By methods similar to the derivation of equation ( 5.1), it can be shown that these translations act on the monodromy coordinates as follows We proceed to check that these formulas are consistent with equation (4.2) in theorem 4.1.…”

Section: Solvable Monodromy Of the Q-okamoto Rational Solutionsmentioning

confidence: 99%

“…The difference equation qP IV (a) is associated to a linear problem (called a Lax pair) [4] Y(qz, t) = A(z; t, f, u)Y(z, t), (…”

Section: Introductionmentioning

confidence: 99%

On symmetric solutions of the fourth q-Painlevé equation

Joshi

Roffelsen

2023

J. Phys. A: Math. Theor.

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The Painlevé equations possess transcendental solutions y(t) with special initial values that are symmetric under rotation or reflection in the complex t-plane. They correspond to monodromy problems that are explicitly solvable in terms of classical special functions. In this paper, we show the existence of such solutions for a q-difference Painlevé equation. We focus on symmetric solutions of a q-difference equation known as qPIV or qP(A5 (1)) and provide their symmetry properties and solve the corresponding monodromy problem.

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“…This idea was proposed in [55]. It is an extension of the ideas presented in [18][19][20]56], where Lax pairs of discrete Painlevé equations are constructed from two-dimensional partial difference equations by using staircase methods.…”

Section: Lax Pairs Of the Qpmentioning

confidence: 99%