On matrix Painlevé hierarchies

We define a matrix first Painlevé hierarchy and a matrix second Painlevé (PIIPII) hierarchy. For our matrix PIIPII hierarchy we also give auto-Bäcklund transformations and consider the iteration of solutions. This is the first paper to define matrix Painlevé hierarchies and to give auto-Bäcklund transformations for a matrix Painlevé 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 PIIPII equation to the obvious candidates for related completely integrable matrix partial differential equations. Our matrix PIIPII hierarchy is placed firmly within the context of a matrix modified Korteweg–de Vries (mKdV) hierarchy. In deriving our matrix PIIPII hierarchy we make use of the Hamiltonian structure of this matrix mKdV hierarchy. We thus see once again the importance for Painlevé hierarchies of the integrability structures of related completely integrable equations.