On the Riemann-Hilbert problem for a $q$-difference Painlevé equation

Abstract

A Riemann-Hilbert problem for a $q$-difference Painlevé equation, known as $q{\text{P}}_{\text{IV}}$, is shown to be solvable. This yields a bijective correspondence between the transcendental solutions of $q{\text{P}}_{\text{IV}}$ and corresponding data on an associated $q$-monodromy surface. We also construct the moduli space of $q{\text{P}}_{\text{IV}}$ explicitly.

32 pages, 1 figure, bibliography updated and 1 reference added