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.