Geometric Aspects of the Painlevé Equations PIII(D<sub>6</sub>) and PIII(D<sub>7</sub>)

Put, Marius van der; Top, Jaap

doi:10.3842/sigma.2014.050

Cited by 22 publications

(12 citation statements)

References 17 publications

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“…But there on both sides of the Riemann-Hilbert correspondence only Zariski open subset were considered. [PT14] builds on [PS09] and constructs a Riemann-Hilbert isomorphism. It has some similarities, but also some differences from our isomorphism.…”

Section: Related Work On Painlevé III and Meromorphic Connectionsmentioning

confidence: 99%

“…(I) (Remarks 9.1 and 9.2 and section 1.5) [JM81] proposes a recipe which relates all solutions of P III (D 6 ) to isomonodromic families of trace free P 3D6 bundles (see also [FIKN06,ch. 5], [PS09], [PT14]). A different recipe, which applies only for the solutions of P III (0, 0, 4, −4), is proposed in [FN80] (see also [IN86], [FIKN06, ch.…”

Section: Open Problemsmentioning

confidence: 99%

“…Second order linear differential equations can be rewritten as first order linear systems in 2 × 2 matrices, and these are equivalent to trivial holomorphic vector bundles of rank 2 on P 1 with meromorphic connections. Such data have been associated to the Painlevé equations in [FN80], [JM81], [IN86], [FIKN06], [PS09], [PT14]. The vector bundle point of view is used in [PS09] and [PT14].…”

Section: Generalities On the Painlevé Equationsmentioning

confidence: 99%

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The Painlevé III equation of type (0,0,4,-4), its associated vector bundles with isomonodromic connections, and the geometry of the movable poles

Guest,

Hertling

2015

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The paper is about a Painlevé III equation of type P III (D 6 ) and its relation to isomonodromic families of vector bundles on P 1 with meromorphic connections.The purpose of the paper is two-fold: it offers a conceptual language for the geometrical objects underlying Painlevé equations, and it offers new results on a particular Painlevé III equation of type P III (D 6 ), which we denote by P III (0, 0, 4, −4). This is equivalent to the radial sine (or sinh) Gordon equation and, as such, it appears very widely in geometry and physics.The length of the paper is due to the systematic development of the material in the language of vector bundles with meromorphic connections, together with their additional structures which take care of all relevant symmetries. Our motivation is to explain in a leisurely fashion the language of this general theory, by means of a concrete example.Complex multi-valued solutions on C * are the natural context for most of the paper, but in the last three chapters real solutions on R >0 (with or without singularities) are addressed. The vector bundles appearing there can be regarded as mixing holomorphic and antiholomorphic structures. They are generalizations of variations of Hodge structures, called TERP structures (for Twistor Extension Real Pairing) and are related to tt * geometry, and harmonic bundles. The paper gives a complete picture of semisimple rank 2 TERP structures. General results on TERP structures can be applied to study the asymptotics near 0 and ∞ of real solutions.Finally, results about the asymptotics of real solutions near 0 and near ∞ are combined with results on the global geometry of the moduli spaces of initial data and monodromy data. This leads to a new global picture of all zeros and poles of all real solutions of P III (0, 0, 4, −4) on R >0 . ContentsChapter 1. Introduction 1.1. A Painlevé equation and its space of initial data M ini 1.2. The space of monodromy data M mon 1.3. Real solutions of P III (0, 0, 4, −4) on R >0 1.4. User's guide to the results 1.5. Related work on Painlevé III and meromorphic connections 1.6. P 3D6 -TEJPA bundles 1.7. TERP(0) bundles, generalizations of Hodge structures 1.8. Open problems 1.9. Acknowledgements Chapter 2. The Riemann-Hilbert correspondence for P 3D6 bundles Chapter 3. (Ir)Reducibility Chapter 4. Isomonodromic families Chapter 5. Useful formulae: three 2 × 2 matrices Chapter 6. P 3D6 -TEP bundles Chapter 7. P 3D6 -TEJPA bundles and moduli spaces of their monodromy tuples Chapter 8. Normal forms of P 3D6 -TEJPA bundles and their moduli spaces Chapter 9. Generalities on the Painlevé equations Chapter 10. Solutions of the Painlevé equation P III (0, 0, 4, −4) Chapter 11. Comparison with the setting of Its, Novokshenov, and Niles Chapter 12. Asymptotics of all solutions near 0 Chapter 13. Rank 2 TEPA bundles with a logarithmic pole 4 CONTENTS Chapter 14. Symmetries of the universal family of solutions of P III (0, 0, 4, −4) Chapter 15. Three families of solutions on R >0 Chapter 16. TERP structures and P 3D6 -TEP bundles...

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Section: Related Work On Painlevé III and Meromorphic Connectionsmentioning

confidence: 99%

Section: Open Problemsmentioning

confidence: 99%

Section: Generalities On the Painlevé Equationsmentioning

confidence: 99%

See 1 more Smart Citation

The Painlevé III equation of type (0,0,4,-4), its associated vector bundles with isomonodromic connections, and the geometry of the movable poles

Guest,

Hertling

2015

Preprint

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“…In the series of papers [17,18,19,21,22] on isomonodromy families for Painlevé equations the cases P I -P IV are treated. Here we apply our methods to degP V , the degenerate fifth Painlevé equation.…”

Section: Introductionmentioning

confidence: 99%

“…Witte [26]. We restrict our transformations to the case of even solutions of degP V and find the classical P V (22), (23) of [26] lead to v 1 = θ 0 + θ 1 and v 2 = θ 0 − θ 1 . Our group of transformations for the even solutions of degP V is generated by (θ 0 , θ 1 ) → (θ 0 + 1, θ 1 ) and (θ 0 , θ 1 ) → (θ 1 , θ 0 ) and the 'trivial' transformations (θ 0 , θ 1 ) → (±θ 0 , ±θ 1 ).…”

mentioning

confidence: 99%

Isomonodromy for the Degenerate Fifth Painlevé Equation

Acosta-Humánez¹,

Put²,

Top

2017

SIGMA

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Abstract. This is a sequel to papers by the last two authors making the Riemann-Hilbert correspondence and isomonodromy explicit. For the degenerate fifth Painlevé equation, the moduli spaces for connections and for monodromy are explicitly computed. It is proven that the extended Riemann-Hilbert morphism is an isomorphism. As a consequence these equations have the Painlevé property and the Okamoto-Painlevé space is identified with a moduli space of connections. Using MAPLE computations, one obtains formulas for the degenerate fifth Painlevé equation, for the Bäcklund transformations.

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