Integrable symplectic maps

The Discrete Korteweg—de Vries Equation

“…It is only relatively recently that integrable maps have been systematically studied (cf. [46][47][48][49][50][51]), for other treatments in the literature. Possibly, this offers a hand to people working in the arena of chaotic phenomena, who now have at their disposal a large number of exactly solvable, yet highly nontrivial, discrete dynamical systems (other than the simple linear ones that were are traditionally used), to study the more analytical aspects of transitions from integrability to chaos.…”

Laurent Polynomials and Superintegrable Maps

“…The situation was much clarified by Veselov [57,58,59] who introduced integrable Lagrange correspondences -a natural discrete-time analogue of Liouville integrable continuous flows -which induce (generically multi-valued) shifts on the associated Liouville tori (see also [3]). Given a continuous integrable system, it is natural to seek a discretization of it that retains both the integrability and as many other properties as possible (e.g.…”

Polynomial integrals for third- and fourth-order ordinary difference equations

“…The discrete nonlinear systems governed by both ordinary difference equations or mappings and partial difference equations or lattice equations have drawn much attention by researchers working under different areas of applied science [1,2,3,5,7,9,10]. Since discrete systems governed by difference equations are more fundamental than the continuous ones described by differential equations their study becomes essential which will lead to the development of a general theory of discrete and in particular nonlinear difference equations.…”

“…Since discrete systems governed by difference equations are more fundamental than the continuous ones described by differential equations their study becomes essential which will lead to the development of a general theory of discrete and in particular nonlinear difference equations. Even though there exists no unique definition of integrability considerable number of analytical methods have been formulated by different groups in recent years to deal with integrability [4,7,8,10,11,12,13,16,18,19,20,21,22,24,25] and significant advancement has already been made for the second order both for autonomous and nonautonomous cases [5,7,9,10,17,19,20,25]. We take the working definition of integrability, here, the one which is related with the existence of sufficient number of integrals of an O∆E.…”