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

Abstract

Abstract. 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 an N-dimensional ellipsoid.

Hirota equation as an example of an integrable symplectic map

“…For related results on symplectic maps see also [9,10]. For a specialist in quantum field theory the quantized Hirota model defines a lattice regularization of a relativistic local field-theoretical model, namely sine-Gordon model.…”

Efficient time-symmetric simulation of torqued rigid bodies using Jacobi elliptic functions

“…In this experiment we perform a comparison also with the DMV methods. We refer to [13] and [15] for a detailed description of these methods, and recall that the higher order DVM are obtained by computing appropriate rescaling of the initial condition. In figure 1 we plot on the x axis the number of floating point operations required by the methods to perform the integration on the interval [0, 1], for different choices of the step size.…”

“…Symplectic integration methods for the Euler equations have been constructed by various authors, [6], [15], [10], [13], see also [9] and references therein. Many of these methods cannot be straightforwardly generalized to the broader class of non canonical Hamiltonian problems, thus their use is limited to the numerical approximation of the FRB equations.…”

“…In this work the authors propose a new implementation of the Discrete Moser-Veselov algorithm of [15], and achieve order four and six in the integration by applying appropriate rescaling of the initial condition. The rescaling need to be performed just once, at the beginning of the integration leading to a significant improvement with respect to the second order Discrete Moser-Veselov algorithm.…”

Isospectral integrability analysis of dynamical systems on discrete manifolds

“…As for different indirect approaches to studying the integrability of differential-difference dynamical systems on discrete manifolds, it is worth mentioning the works [19][20][21][22][23]35] based on the inverse spectral transform and related Lie-algebraic methods, where a priori Lax integrable Hamiltonian flows possessing infinite hierarchies of conservation laws are constructed. Many important analytical properties of these other approaches were constructively incorporated into the algorithmic gradient-holonomic scheme presented above.…”