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In this paper, we discuss several concepts of the modern theory of discrete integrable systems, including: • Time discretization based on the notion of Bäcklund transformation. • Symplectic realizations of multi-Hamiltonian structures. • Interrelations between discrete 1D systems and lattice 2D systems. • Multi-dimensional consistency as integrability of discrete systems. • Interrelations between integrable systems of quad-equations and integrable systems of Laplace type. • Pluri-Lagrangian structure as integrability of discrete variational systems. All these concepts are illustrated by the discrete time Toda lattices and their relativistic analogs. [ABSTRACT FROM AUTHOR]

We establish the pluri-Lagrangian structure for families of Bäcklund transformations of relativistic Toda-type systems. The key idea is a novel embedding of these discrete-time (one-dimensional) systems into certain two-dimensional (2D) 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 Bäcklund transformations. These superposition formulae, in turn, are key ingredients necessary to understand and to prove commutativity of the multi-valued Bäcklund transformations. Furthermore, we discover a 2D generalization of the spectrality property known for families of Bäcklund transformations. This result produces a family of local conservations laws for 2D pluri-Lagrangian lattice systems, with densities being derivatives of the discrete 2-form with respect to the Bäcklund (spectral) parameter. Thus, a relation of the pluri-Lagrangian structure with more traditional integrability notions is established. [ABSTRACT FROM AUTHOR]