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Discrete versions of some classical integrable systems and factorization of matrix polynomials

1991; Springer Science+Business Media; Volume: 139; Issue: 2 Linguagem: Inglês

10.1007/bf02352494

1432-0916

Jürgen Moser, А. П. Веселов,

Nonlinear Photonic Systems

Discrete versions of several classical integrable systems are investigated, such as a discrete analogue of the higher dimensional force-free spinning top (Euler-Arnold equations), the Heisenberg chain with classical spins and a new discrete system on the Stiefel manifold. The integrability is shown with the help of a Lax-pair representation which is found via a factorization of certain matrix polynomials. The complete description of the dynamics is given in terms of Abelian functions; the flow becomes linear on a Prym variety corresponding to a spectral curve. The approach is also applied to the billiard problem in the interior of anN-dimensional ellipsoid.