Bäcklund transformations for the elliptic Gaudin model and a Clebsch system

Abstract

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.

Algebraic entropy for algebraic maps

“…This immediately implies that the algebraic entropy is the logarithm of the golden mean, in agreement with the value (19) found for the corresponding algebraic map (11). The map (20) appears in various other contexts. For a = 3 it can be used to generate sequences of Markoff numbers, which appear in Diophantine approximation theory.…”

“…A clue in this direction is given by the fact that Bäcklund transformations, when expressed in uniformizing variables, represent addition formulae for the corresponding functions [21]. The first example we considered is a non-integrable algebraic map, namely (11), which arises from the birational map (20) in one dimension higher, by restricting to a level set of its polynomial first integral. It is interesting to observe that the algebraic entropy of the birational map is the same as that for the algebraic map obtained via this restriction, and it would be worth examining other examples of this kind to see if this is a general phenomenon.…”

Bäcklund transformations and Hamiltonian flows

“…Then we can start to iterate the map according to (19). Thanks to Theorem 2, the result will be a sampling of the particular trajectory of the continuous flow (20) corresponding to the same initial conditions. Note that the functions c k depend only on the conserved quantities and on the parameters of the transformations, so they are constants along the trajectories of the model, i.e.…”

On an integrable discretisation of the Ablowitz-Ladik hierarchy

“…However, he seems to not recognize that the algebraic equations solved by certain particular combinations of the dynamical variables appearing in the matrix D k (called "auxiliary variables" in [10]), can be found by evaluating the determinant of the dressing matrix at a certain constant value of λ and then setting the result equal to zero. The relevance of this fact comes from the observation that it allows to obtain explicit transformations, as shown for example in [7], [8], [11] [12]. This may imply an improvement in the computations of the discretisation scheme.…”