The Problem of Integrable Discretization: Hamiltonian Approach

Leading order integrability conditions for differential-difference equations

“…Most of this literature is focused on the construction of integrable systems. For a symmetry approach see the work of Yamilov and his coworkers [1,12,13,17] and papers cited therein, [14] for an exhaustive discussion of the Hamiltonian approach and [15] for Jacobi operator approach. In contrast, this paper examines the integrability or lack of integrability of a lattice directly.…”

“…Their integrability properties are also of interest in their own right. See, for example, [14,15]. Of particular interest are conservation laws for (1.1) which depend on arbitrarily many shifts of the dynamical varible, w q .…”

Functional Representation of the Volterra Hierarchy

“…To my knowledge this is a new result. Of course, it is closely related to the discrete time approach which has been developed for the Volterra model by different authors (see book of Suris [12] and references therein). However equations discussed here were considered not as discrete but as ones with analytic dependence on the deformation parameter and, secondly, can be viewed as a realization of the discrete maps in terms of the usual Volterra flows.…”

A Nonlocal Kac-van Moerbeke Equation Admitting N-Soliton Solutions

“…(1.6) (See also Suris' forthcoming monograph Ref. [10] for an up-to-date and comprehensive account of these lattice systems and a host of other ones.) Little appears to be known concerning the general theory of nonlocal evolution equations such as (1.2)-(1.4).…”