The Second Painlevé Hierarchy and the Stationary KdV Hierarchy

Abstract

It is well known that soliton equations such as the Korteweg–de Vries equation are members of infinite sequences of PDEs known as hierarchies. Here we consider infinite sequences of ODEs associated with the Painlevé equations. We review methods of constructing such hierarchies, specifically the second Painlevé hierarchy, as reductions of PDE hierarchies. We also show that in the large-parameter limit, the solutions of the second Painlevé hierarchy are given by the periodic solutions of the stationary KdV hierarchy.