Auto-Backlund transformations and special integrals for differential-delay Painlevé hierarchies

Abstract

The six Painleve equations have attracted much interest over the last thirty years or so. More recently many authors have begun to explore properties of higher-order versions of both these equations and their discrete analogues. However, little attention has been paid to differential-delay Painleve equations, i.e., analogues of the Painleve equations involving both shifts in and derivatives with respect to the independent variable, and even less to higher-order analogues of these last. In the current paper we discuss the phenomenon whereby members of one differential-delay Painleve hierarchy define solutions of higher-order members of a second differential-delay Painleve hierarchy. We also give an auto-Backlund transformation for a differential-delay Painleve hierarchy. The key to our approach is the underlying Hamiltonian structure of related completely integrable lattice hierarchies. (C) 2014 Elsevier B.V. All rights reserved.