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

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 $q{P}_{\mathrm{VI}}$ admits the differential Okamoto space \emph{via} some natural limit process.

References in corpus (3)

• Geometric Aspects of Painlevé Equations
• Building meromorphic solutions of $q$-difference equations using a Borel-Laplace summation
• Isomonodromic deformation of q-difference equations and confluence