Bäcklund transformations as exact integrable time discretizations for the trigonometric Gaudin model

Abstract

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.

The Gross–Pitaevskii equation: Bäcklund transformations and admitted solutions

“…The results here presented are part of a long time investigation program which covers various aspects of the study on nonlinear evolution equations and their structural properties as well as the construction of solutions admitted by particular problems (sse e.g. [6,3], [16,17,18], [26,27]). Here we applied the transformations found in [4] to the Gross-Pitaevskii equation in 1+1 dimensions.…”

A q-difference Baxter operator for the Ablowitz–Ladik chain

“…Now we shall show how it is possible to obtain the classical version of the Baxter's equation (21) by looking at the action of the dressing matrix on the matrices L k (λ). The result will be trivial, but it will help to understand what we get in the quantum case.…”

“…The idea is a mix of the observations due to Pasquier & Gaudin [17] and Kuznetsov & Sklyanin [14]. The Baxter's equation (21) involves the trace of the monodromy matrix so we would like to obtain an expression for this trace. The observation in [17] is that TrL(µ) doesn't change if we perform a sort of similarity transformation on the matrices L k likeL k = M −1 k+1 L k M k .…”

Bäcklund Transformations for the Kirchhoff Top

“…As shown in [9], to ensure "reality" of the maps (3.4), one has to require the Darboux matrix D to be a unitary matrix (possibly up to an irrelevant scalar factor); this holds true iff λ ± are mutually complex conjugate, i.e. iff λ 0 is real and µ is pure imaginary.…”

“…Indeed plays the role of time step for the one parameter (λ 0 ) discrete dynamics defined by the Bäcklund transformations. By following [9], in order to identify the continuous limit of this discrete dynamics we take the Taylor expansion of the dressing matrix at order , obtaining:…”