On matrix Painleve hierarchies

We define a matrix first Painleve hierarchy and a matrix second Painleve (P-II) hierarchy. For our matrix P-II hierarchy we also give auto-Backlund transformations and consider the iteration of solutions. This is the first paper to define matrix Painleve hierarchies and to give auto-Backlund transformations for a matrix Painleve hierarchy. We also consider, amongst other results, the derivation of sequences of special integrals and autonomous limits. Until now it has been unknown how to connect the known matrix P-II equation to the obvious candidates for related completely integrable matrix partial differential equations. Our matrix P-II hierarchy is placed firmly within the context of a matrix modified Korteweg-de Vries (mKdV) hierarchy. In deriving our matrix P-II hierarchy we make use of the Hamiltonian structure of this matrix mKdV hierarchy. We thus see once again the importance for Painleve hierarchies of the integrability structures of related completely integrable equations. (C) 2016 Elsevier Inc. All rights reserved.