Auto-Bäcklund Transformations for New Matrix First and Second Painlevé Hierarchies

Abstract

We define a new doubly extended matrix second Painlevé hierarchy, and in addition a new extended matrix first Painlevé hierarchy. For the former, we present three auto-Bäcklund transformations (auto-BTs) that constitute nontrivial extensions to our new hierarchy of previously derived results on the auto-BTs of a much simpler matrix second Painlevé hierarchy; for the latter, we define an auto-BT analagous to the third of these matrix second Painlevé auto-BTs. In addition, for both of our new hierarchies, we define a new class of auto-BT. In the scalar reduction, these then give rise to a new Bäcklund process for the second Painlevé equation, as well as to a similar Bäcklund process for the first Painlevé equation. These new Bäcklund processes provide mappings of arbitrary constants appearing in solutions.