Discrete Painlevé equations and their Lax pairs as reductions of integrable lattice equations

Abstract

We describe a method to obtain Lax pairs for periodic reductions of a rather general class of integrable non-autonomous lattice equations. The method is applied to obtain reductions of the non-autonomous discrete Korteweg–de Vries equation and non-autonomous discrete Schwarzian Korteweg–de Vries equation, which yield a discrete analogue of the fourth Painlevé equation, a q-analogue of the sixth Painlevé equation and the q-Painlevé equation with a symmetry group of affine Weyl type E(1)6.