The Hamiltonian Structure of the Second Painleve Hierarchy

Abstract

In this paper we study the Hamiltonian structure of the second Painleve 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 ${\alpha }_n$. 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.

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Cited by in corpus (16)

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• A matrix model with a singular weight and Painleve' III
• Total integrals of global solutions to Painleve II
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• Hamiltonian structure of rational isomonodromic deformation systems
• Solutions of the Painleve VI Equation from Reduction of Integrable Hierarchy in a Grassmannian Approach
• Double scaling limits of random matrices and minimal (2m,1) models: the merging of two cuts in a degenerate case
• Non-autonomous reductions of the KdV equation and multi-component analogs of the Painlevé equations P${}_{34}$ and P${}_3$
• Non-Integrability of Some Higher-Order Painlevé Equations in the Sense of Liouville
• Rational Solutions of the Sasano System of Type $A_5^{(2)}$
• The Sigma Form for the PII Hierarchy
• On symmetries of the non-stationary PIIn hierarchy and their applications
• Asymptotics of the deformed higher order Airy-kernel determinants and applications
• Asymptotic Properties of a Special Solution to the (3,4) String Equation
• Meromorphy of solutions for a wide class of ordinary differential equations of Painlevé type
• A Lagrangian Description of the Higher-Order Painlevé Equations