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 n-th element of the hierarchy is a non linear ODE of order 2n in the independent variable z depending on n parameters denoted by t 1 , . . . , t n−1 and αn. We introduce new canonical coordinates and obtain Hamiltonians for the z and t 1 , . . . , t n−1 evolutions. We give explicit formulae for these Hamiltonians showing that they are polynomials in our canonical coordinates.1991 Mathematics Subject Classification. Primary 34M55. Secondary 37K20, 35Q53.

Higher‐order analogues of the Tracy‐Widom distribution and the Painlevé II hierarchy

“…This Lax system was found by Flaschka and Newell in [22] for the Painlevé II equation, and generalized to the Painlevé II hierarchy in [13,34,38].…”

Hamiltonian Structure of PI Hierarchy

“…A similar idea, however, can be found in the recent work of Mazzocco and Mo [41] on an isomonodromic hierarchy related to the second Painlevé equation. They start from the LiePoisson structure of a loop algebra, and convert the Hamiltonian structure on a coadjoint orbit to a Hamiltonian system in Darboux coordinates by a time-dependent canonical transformation.…”

The generalized Yablonskii–Vorob'ev polynomials and their properties

“…Interestingly, that solutions of the Nth equation in the generalized second Painlevé hierarchy with N > 1 also satisfy the equations (2m + 1)∂ t m w + ∂ z (∂ z + 2w)L m ∂ z w − w 2 = 0, 1 m N − 1, (5.1) which belong to a rescaled version of the mKdV hierarchy [1]. This, in particular, means that rational solutions of the mKdV (and, consequently, KdV) equation are directly expressible as logarithmic derivatives of the generalized Yablonskii-Vorob'ev polynomials.…”

“…The hierarchy appears as similarity reduction of the modified Korteweg-de Vries (mKdV) hierarchy and possesses several mathematical and physical applications [1,2]. The aim of this Letter is to derive special polynomials V (N) n (z; t 1 , .…”

“…referred to as the Yablonskii-Vorob'ev polynomials [5,6]. In our designations Yablonskii and Vorob'ev dealt with the polynomials V (1) n (z; 1). Analogous special polynomials for the equations in the hierarchy with t 1 = · · · = t N−1 = 0, t N = 1 were introduced in [7] and studied in [8,9].…”