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The Hamiltonian Structure of the Second Painleve Hierarchy

· 2006 · nlin.SI · arXiv nlin/0610066

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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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A non-commutative discrete first Painlev\'e hierarchy: the Lax pair approach

nlin.SI · 2026-05-28 · unverdicted · novelty 6.0

Constructs non-commutative discrete first Painlevé hierarchy d-PI_m^nc via non-commutative isomonodromic problem, expresses both commutative and non-commutative versions via Svinin polynomials, derives reduction from non-commutative Volterra lattice, and gives continuous limits for first three membe

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  • A non-commutative discrete first Painlev\'e hierarchy: the Lax pair approach nlin.SI · 2026-05-28 · unverdicted · none · ref 8 · internal anchor

    Constructs non-commutative discrete first Painlevé hierarchy d-PI_m^nc via non-commutative isomonodromic problem, expresses both commutative and non-commutative versions via Svinin polynomials, derives reduction from non-commutative Volterra lattice, and gives continuous limits for first three membe