Three-dimensional discrete systems of Hirota-Kimura type and deformed Lie-Poisson algebras

TL;DR

This paper develops birational discretizations of algebraically integrable systems related to Lie-Poisson algebras, demonstrating their bi-Hamiltonian structure and integrability, extending the Hirota-Kimura discretization approach to new classes of systems.

Contribution

It constructs new discretizations of Lie-Poisson algebra systems that preserve bi-Hamiltonian structure and integrability, generalizing the Hirota-Kimura scheme.

Findings

Discretizations are bi-Hamiltonian with compatible Poisson brackets.

Constructed maps are one-parameter deformations of original Lie-Poisson algebras.

All maps satisfy Diophantine integrability criteria.

Abstract

Recently Hirota and Kimura presented a new discretization of the Euler top with several remarkable properties. In particular this discretization shares with the original continuous system the feature that it is an algebraically completely integrable bi-Hamiltonian system in three dimensions. The Hirota-Kimura discretization scheme turns out to be equivalent to an approach to numerical integration of quadratic vector fields that was introduced by Kahan, who applied it to the two-dimensional Lotka-Volterra system. The Euler top is naturally written in terms of the $\mathfrak{so}(3)$ Lie-Poisson algebra. Here we consider algebraically integrable systems that are associated with pairs of Lie-Poisson algebras in three dimensions, as presented by G\"umral and Nutku, and construct birational maps that discretize them according to the scheme of Kahan and Hirota-Kimura. We show that the maps thus obtained are also bi-Hamiltonian, with pairs of compatible Poisson brackets that are one-parameter deformations of the original Lie-Poisson algebras, and hence they are completely integrable. For comparison, we also present analogous discretizations for three bi-Hamiltonian systems that have a transcendental invariant, and finally we analyze all of the maps obtained from the viewpoint of Halburd's Diophantine integrability criterion.