A symmetric difference-differential Lax pair for Painlev\'e VI

Christopher M. Ormerod; Eric M. Rains

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A symmetric difference-differential Lax pair for Painlev\'e VI

Christopher M. Ormerod, Eric M. Rains

Exactly Solvable and Integrable Systems · Submitted 14 Mar 2016

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We present a Lax pair for the sixth Painlev\'e equation arising as a continuous isomonodromic deformation of a system of linear difference equations with an additional symmetry structure.

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What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.06 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.09 · mentions code or data availability 0.02.

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We present a Lax pair for the sixth Painlev\'e equation arising as a continuous isomonodromic deformation of a system of linear difference equations with an additional symmetry structure. We call this a symmetric difference-differential Lax pair. We show how the discrete isomonodromic deformations of the associated linear problem gives us a discrete version of the fifth Painlev\'e equation. By considering degenerations we obtain symmetric difference-differential Lax pairs for the fifth Painlev\'e equation and the various degenerate versions of the third Painlev\'e equation.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We present a Lax pair for the sixth Painlev\'e equation arising as a continuous isomonodromic deformation of a system of linear difference equations with an additional symmetry structure.	contribution 0.99	0.94	0.73	asserted 1.00
s1	We call this a symmetric difference-differential Lax pair.	contribution 0.66	0.72	0.16	asserted 0.97
s2	We show how the discrete isomonodromic deformations of the associated linear problem gives us a discrete version of the fifth Painlev\'e equation.	result 0.86	0.94	0.14	asserted 1.00
s3	By considering degenerations we obtain symmetric difference-differential Lax pairs for the fifth Painlev\'e equation and the various degenerate versions of the third Painlev\'e equation.	result 0.67	0.94	0.21	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.87 · evidence from proof or derivation 1.00.

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Original abstract

We present a Lax pair for the sixth Painlev\'e equation arising as a continuous isomonodromic deformation of a system of linear difference equations with an additional symmetry structure. We call this a symmetric difference-differential Lax pair. We show how the discrete isomonodromic deformations of the associated linear problem gives us a discrete version of the fifth Painlev\'e equation. By considering degenerations we obtain symmetric difference-differential Lax pairs for the fifth Painlev\'e equation and the various degenerate versions of the third Painlev\'e equation.

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2 arXiv papers cite this in 3 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

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An important property of two of the systems introduced was that the solutions satisfied an additional symmetry property, which we can generalize (see [ref] ) to systems of the form [ref] satifying the additional property [equation] in the case $σ= σ_h$ and [equation] in the case in which $σ= σ_q$.

An elliptic Garnier system · Introduction · 0.93

Cites it as background 1 paper

It has also been reported that a compatibility condition of linear difference equation and its deformation equation also give a Painlevé equation [ref] .

Reduction of lattice equations to the Painlev\'e equations: P$_{\rm IV}$ and P$_{\rm V}$ · Background · 0.81

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Cites as background 1 paper

These relaxations of the difference equations and their moduli spaces have been the subject of a recent study [ref] .

Generalized Hitchin systems on rational surfaces · Background and motivation · 0.96

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