Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Degeneration from difference to differential Okamoto spaces for the sixth Painlevé equation

Thomas Dreyfus, Viktoria Heu

Abstract

In the current paper we study the $q$-analogue introduced by Jimbo and Sakai of the well known Painlevé VI differential equation. We explain how it can be deduced from a $q$-analogue of Schlesinger equations and show that for a convenient change of variables and auxiliary parameters, it admits a $q$-analogue of Hamiltonian formulation. This allows us to show that Sakai's $q$-analogue of Okamoto space of initial conditions for $qP_\mathrm{VI}$ admits the differential Okamoto space \emph{via} some natural limit process.

Degeneration from difference to differential Okamoto spaces for the sixth Painlevé equation

Abstract

In the current paper we study the -analogue introduced by Jimbo and Sakai of the well known Painlevé VI differential equation. We explain how it can be deduced from a -analogue of Schlesinger equations and show that for a convenient change of variables and auxiliary parameters, it admits a -analogue of Hamiltonian formulation. This allows us to show that Sakai's -analogue of Okamoto space of initial conditions for admits the differential Okamoto space \emph{via} some natural limit process.

Paper Structure

This paper contains 22 sections, 24 theorems, 264 equations.

Table of Contents

  1. Fuchsian systems
  2. Differential case
  3. A discrete analogue
  4. Confluence
  5. Schlesinger equations
  6. Differential case
  7. A discrete analogue
  8. Confluence
  9. The $q$-Schlesinger equations discretize the differential ones
  10. The $q$-Lax pairs discretize the differential ones
  11. The difference equations for triples $(\boldsymbol{\lambda}, \boldsymbol{y},\boldsymbol{Z})$ discretize the differential ones
  12. The sixth Painlevé equation
  13. Differential case
  14. A discrete analogue
  15. Confluence
  16. ...and 7 more sections

Key Result

Lemma 1.2

If a family of $\mathfrak{sl}_2$-Fuchsian systems sl2System with spectral data $\boldsymbol{\theta}$, with $\theta_\infty \neq 0$, gives rise to as above, then the coefficients of the matrix $A$ necessarily are the following functions of $\lambda,y,Z, x,t$ and $\boldsymbol{\theta}$: Here we denote

Theorems & Definitions (63)

  • Definition 1.1
  • Lemma 1.2
  • proof
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Definition 1.6
  • Remark 1.7
  • Remark 1.8
  • Lemma 1.9
  • ...and 53 more