In this work we show that, under certain conditions, parametric Bäcklund transformations (BTs) for a finite dimensional integrable system can be interpreted as solutions to the equations of motion defined by an associated non-autonomous Hamiltonian. The two systems share the same constants of motion. This observation leads to the identification of the Hamiltonian interpolating the iteration of the discrete map defined by the transformations, that indeed in numerical applications can be considered a linear combination of the integrals appearing in the spectral curve of the Lax matrix. An example with the periodic Toda lattice is given.
We construct a two-parameter family of Bäcklund transformations for the trigonometric classical Gaudin magnet. The approach follows closely the one introduced by E. V.Kuznetsov (1998,1999) in a number of seminal papers, and takes advantage of the intimate relation between the trigonometric and the rational case. As in the paper by A.Hone, V.Kuznetsov and one of the authors (O.R.) (2001) the Bäcklund transformations are presented as explicit symplectic maps, starting from their Lax representation. The (expected) connection with the xxz Heisenberg chain is established and the rational (xxx ) case is recovered in a suitable limit. It is shown how to obtain a "physical" transformation mapping real variables into real variables. The interpolating Hamiltonian flow is derived and some numerical iterations of the map are presented.
Following the general results on the relationships about Bäcklund transformations (BTs) and exact discretisation given in a previous work [12], we consider the Ablowitz-Ladik hierarchy and a corresponding family of BTs. After discussing the boundary conditions, we show how to get explicit transformations. The Hamiltonian properties of the maps and of the discrete flows are examined. The conditions on the parameters of the map giving exact discretisations are discussed. Finally, analytical and numerical examples are given.
A two-parameters family of Bäcklund transformations for the classical elliptic Gaudin model is constructed. The maps are explicit, symplectic, preserve the same integrals as for the continuous flows and are a time discretization of each of these flows. The transformations can map real variables into real variables, sending physical solutions of the equations of motion into physical solutions. The starting point of the analysis is the integrability structure of the model. It is shown how the analogue transformations for the rational and trigonometric Gaudin model are a limiting case of this one. An application to a particular case of the Clebsch system is given.
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