On the monodromy manifold of $q$-Painlev\'e VI and its Riemann-Hilbert problem
Abstract
We study the sixth -difference Painlev\'e equation () through its associated Riemann-Hilbert problem (RHP) and show that the RHP is always solvable for irreducible monodromy data. This enables us to identify the solution space of with a monodromy manifold for generic parameter values. We deduce this manifold explicitly and show it is a smooth and affine algebraic surface when it does not contain reducible monodromy. Furthermore, we describe the RHP for reducible monodromy data and show that, when solvable, its solution is given explicitly in terms of certain orthogonal polynomials yielding special function solutions of .
Keywords
Cite
@article{arxiv.2202.10597,
title = {On the monodromy manifold of $q$-Painlev\'e VI and its Riemann-Hilbert problem},
author = {Nalini Joshi and Pieter Roffelsen},
journal= {arXiv preprint arXiv:2202.10597},
year = {2023}
}
Comments
The affine Segre surface defined in equations (2.26) has been rescaled such that it is invariant under translations of the parameters. Details of the computation of the determinant of the Hessian matrix in the proof of Proposition 5.5 have been added. Three remarks have been added, Remarks 2.4, 2.23 and 5.6