Integrable symplectic maps (Q810962)

The paper first reviews the concept of a symplectic map, its characterization and properties and various ways of construction. Among these is a rather formal discrete version of Euler-Lagrange equations, which relates to earlier work by Maeda on discrete Hamiltonian and Lagrangian systems. Integrability of a symplectic map essentially means linearizability, and an analogue of the Arnold-Liouville theorem states that such integrability follows from the existence of a sufficient number of independent invariants in involution. The second part of the paper shows how integrable symplectic maps can be obtained from stationary solutions of discrete versions of known integrable evolution equations, such as the non-linear Schrödinger equation and the KdV equation.