The Hamiltonian structure of the second Painlevé hierarchy

Abstract

In this paper we study the Hamiltonian structure of the second Painlevé hierarchy, an infinite sequence of nonlinear ordinary differential equations containing PII as its simplest equation. The nth element of the hierarchy is a nonlinear ODE of order 2n in the independent variable z depending on n parameters denoted by t1, …, tn−1 and αn. We introduce new canonical coordinates and obtain Hamiltonians for the z and t1, …, tn−1 evolutions. We give explicit formulae for these Hamiltonians showing that they are polynomials in our canonical coordinates.