Chapter 17. On the Gauge Transformation of the Sixth Painlevé Equation

Sasaki, Yoshikatsu

doi:10.1515/9783110275667.137

Search citation statements

Order By: Relevance

Paper Sections

Select...

Painlevé VI and Hamiltonian System On The Moduli Space1

Introduction1

Citation Types

Supporting

Mentioning

Contrasting

Year Published

2016

Publication Types

Select...

Article1

Relationship

Self Cite0

Independent1

Authors

Journals

Cited by 1 publication

(2 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…By Theorem 2.1, it suffices to prove the sufficient part. Suppose there exists a single-valued solution Ω 12 (z; τ ) to (2.25) 25), it is easy to see that Ω (z; τ ) satisfies the integrability condition (2.15) (see e.g. [15, Proposition 3.5.1]), namely (2.14) is completely integrable.…”

Section: Painlevé VI and Hamiltonian System On The Moduli Spacementioning

confidence: 99%

See 1 more Smart Citation

Hamiltonian system for the elliptic form of Painlevé VI equation

Chen

Kuo

Lin

2016

Journal de Mathématiques Pures et Appliquées

View full text Add to dashboard Cite

In literature, it is known that any solution of Painlevé VI equation governs the isomonodromic deformation of a second order linear Fuchsian ODE on CP 1 . In this paper, we extend this isomonodromy theory on CP 1 to the moduli space of elliptic curves by studying the isomonodromic deformation of the generalized Lamé equation. Among other things, we prove that the isomonodromic equation is a new Hamiltonian system, which is equivalent to the elliptic form of Painlevé VI equation for generic parameters. For Painlevé VI equation with some special parameters, the isomonodromy theory of the generalized Lamé equation greatly simplifies the computation of the monodromy group in CP 1 . This is one of the advantages of the elliptic form. u zz − 1 2 u 2 z z = (u zz ) z − u z u zz = − 1 4 e u z + 1 4 e u u z = 0.

show abstract

Section: Painlevé VI and Hamiltonian System On The Moduli Spacementioning

confidence: 99%

“…The aim is to classify those nonlinear ODEs whose solutions have the so-called Painlevé property. We refer the reader to [1,5,6,9,10,11,12,13,15,17,18,19,20,21,22,24,25] and references therein for some historic account and the recent developments. Painlevé VI with four free parameters (α, β, γ, δ) can be written as…”

Section: Introductionmentioning

confidence: 99%