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The splitting approach of the previous paper by Suris (Suris Yu B 1997 Inverse Problems 13 1121-36) is applied to the original Ablowitz-Ladik discretization of nonlinear Schrodinger equations (NLS). The outcome is the realization of the latter free of nonlocalities. The severe grid-size restriction pointed out in the comment by Black et al (Black W, Weidemann J A C and Herbst B M 1999 Inverse Problems 15 807-10) does not appear in this construction.

We discuss some of the integrable lattices introduced recently by R Yamilov. We demonstrate that they are closely related to the usual Toda lattice by means of a sort of Backlund transformations. We also apply the general procedure of integrable discretization and obtain their integrable finite-difference approximations. These novel integrable discrete-time systems are also related to the discrete-time Toda lattice by means of the Backlund transformations. The whole construction exploits the tri-Hamiltonian structure of the Toda lattice.

We propose a new method for studying asymptotic behaviour of complex separatrices of some symplectic maps. Our approach gives an alternative method (as opposed to iterating the maps) for computing some constants to measure the splitting of separatrices. Details are presented for semistandard and cubic standard-like maps.

We establish the pluri-Lagrangian structure for families of B\"acklund transformations of relativistic Toda-type systems. The key idea is a novel embedding of these discrete-time (one-dimensional) systems into certain two-dimensional pluri-Lagrangian lattice systems. This embedding allows us to identify the corner equations (which are the main building blocks of the multi-time Euler-Lagrange equations) with local superposition formulae for B\"acklund transformations. These superposition formulae, in turn, are key ingredients necessary to understand and to prove commutativity of the multi-valued B\"acklund transformations. Furthermore, we discover a two-dimensional generalization of the spectrality property known for families of B\"acklund transformations. This result produces a family of local conservations laws for two-dimensional pluri-Lagrangian lattice systems, with densities being derivatives of the discrete 2-form with respect to the B\"acklund (spectral) parameter. Thus, a relation of the pluri-Lagrangian structure with more traditional integrability notions is established., Comment: 25 pages

General Lagrangian theory of discrete one-dimensional integrable systems is illustrated by a detailed study of B\"acklund transformations for Toda-type systems. Commutativity of B\"acklund transformations is shown to be equivalent to consistency of the system of discrete multi-time Euler-Lagrange equations. The precise meaning of the commutativity in the periodic case, when all maps are double-valued, is established. It is shown that gluing of different branches is governed by the so called superposition formulas. The closure relation for the multi-time Lagrangian 1-form on solutions of the variational equations is proved for all Toda-type systems. Superposition formulas are instrumental for this proof. The closure relation was previously shown to be equivalent to the spectrality property of B\"acklund transformations, i.e., to the fact that the derivative of the Lagrangian with respect to the spectral parameter is a common integral of motion of the family of B\"acklund transformations. We relate this integral of motion to the monodromy matrix of the zero curvature representation which is derived directly from equations of motion in an algorithmic way. This serves as a further evidence in favor of the idea that B\"acklund transformations serve as zero curvature representations for themselves., Comment: 27 pp