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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 Backlund 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.
We show that Toda shock waves are asymptotically close to a modulated finite gap solution in the region separating the soliton and the elliptic wave regions. We previously derived formulas for the leading terms of the asymptotic expansion of these sh
The time evolution problem for non-self adjoint second order differential operators is studied by means of the path integral formulation. Explicit computation of the path integral via the use of certain underlying stochastic differential equations, w
A connection between the finite ultradiscrete Toda lattice and the box-ball system is extended to the case where each box has own capacity and a carrier has a capacity parameter depending on time. In order to consider this connection, new carrier rul
A q-discrete version of the two-dimensional Toda molecule equation is proposed through the direct method. Its solution, Backlund transformation and Lax pair are discussed. The reduction to the q-discrete cylindrical Toda molecule equation is also discussed.
We establish the Lagrangian nature of the discrete isospectral and isomonodromic dynamical systems corresponding to the re-factorization transformations of the rational matrix functions on the Riemann sphere. Specifically, in the isospectral case we