The Hamiltonian Structure of the Second Painleve Hierarchy

Details The Hamiltonian Structure of the Second Painleve Hierarchy The Hamiltonian Structure of the Second Painleve Hierarchy

2006-10-27

arxiv.org/abs/nlin/0610066v1

Nonlinear Sciences

Exactly Solvable and Integrable Systems

Scientific paper

10.1088/0951-7715/20/12/006

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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