New results on integrability of the Kahan-Hirota-Kimura discretizations
For researchers in integrable systems and discrete geometry, this work provides explicit algebraic formulas and computational evidence for integrability of discretizations, but is incremental as it extends known results.
The paper reports new observations on the integrability of Kahan-Hirota-Kimura discretizations for several complex cases (Clebsch system, Kirchhoff system, Lagrange top), providing compact formulas for integrals of motion and invariant measure densities, and establishing higher-order Wronskian Hirota-Kimura bases via computer algebra.
R. Hirota and K. Kimura discovered integrable discretizations of the Euler and the Lagrange tops, given by birational maps. Their method is a specialization to the integrable context of a general discretization scheme introduced by W. Kahan and applicable to any vector field with a quadratic dependence on phase variables. We report several novel observations regarding integrability of the Kahan-Hirota-Kimura discretization. For several of the most complicated cases for which integrability is known (Clebsch system, Kirchhoff system, and Lagrange top), - we give nice compact formulas for some of the more complicated integrals of motion and for the density of the invariant measure, and - we establish the existence of higher order Wronskian Hirota-Kimura bases, generating the full set of integrals of motion. While the first set of results admits nice algebraic proofs, the second one relies on computer algebra.