Elements of the integrability theory of discrete dynamical systems (Q1097418)

It is well known that a nonlinear dynamical system \(u_ t=K[u]\) with Lax type representation has on an infinite dimensional smooth manifold \(M\ni u\) two symplectic structures \(\{\cdot,\cdot \}_{{\mathcal L}}\) and \(\{\cdot,\cdot \}_{{\mathcal M}}\), which are compatible, that is the symplectic structure for the sum (\({\mathcal L}+\lambda {\mathcal M})\) is conserved for all \(\lambda \in {\mathbb{R}}^ 1\), where the operators \({\mathcal L},{\mathcal M}: M\to T(M)\) are implectic by definition. The inverse problem, i.e. to find the implectic pair (\({\mathcal L},{\mathcal M})\) for the dynamical system \(u_ t=K[u]\), if they exist, and also the problem of construction of conservation laws in exact form were algorithmically solved in the papers by \textit{Yu. A. Mitropolskij}, the author and \textit{V. Gr. Samojlenko} [Sov. Math. Dokl. 33, 542-546 (1986); translation from Dokl. Akad. SSSR 287, 1312-1317 (1986; Zbl 0614.58024)]. In the context of the above cited papers the paper under review is concerned also with the problem (inverse) of integrability of discrete dynamical systems on the discrete manifold \(M\simeq {\mathbb{R}}^{{\mathbb{Z}}}\). On this manifold also the differential geometrical structure is built. Having used this geometrical structure namely, the discrete Grassmann algebra complex, the Lagrangian structure of a discrete dynamical system in the case of integrability is investigated in detail. In particular, the discrete analog of the Novikov equations and their complete integrability by Liouville are stated on the finite zone submanifold of M. As an example, the discrete nonlinear dynamical system of Schrödinger is examined. It must be mentioned that a differential-geometric part of the paper under review is close to the analoguous results by \textit{B. A. Kupershmidt} [Astérisque 123, 1-58 (1985; Zbl 0565.58024)] which were obtained independently.