Non-commutative q-Painlevé VI equation

Abstract

By applying suitable centrality conditions to non-commutative non-isospectral lattice modified Gel'fand–Dikii type systems we obtain the corresponding non-autonomous equations. Then we derive the non-commutative q-discrete Painlevé VI equation with full range of parameters as the (2, 2) similarity reduction of the non-commutative, non-isospectral and non-autonomous lattice modified Korteweg–de Vries equation. We also comment on the fact that in making the analogous reduction starting from the Schwarzian Korteweg–de Vries equation no such ‘non-isospectral generalization’ is needed.