From su(2) Gaudin Models to Integrable Tops

In the present paper we derive two well-known integrable cases of rigid body dynamics (the Lagrange top and the Clebsch system) performing an algebraic contraction on the two-body Lax matrices governing the (classical) su(2) Gaudin models. The procedure preserves the linear r-matrix formulation of the ancestor models. We give the Lax representation of the resulting integrable systems in terms of su(2) Lax matrices with rational and elliptic dependencies on the spectral parameter. We finally give some results about the many-body extensions of the constructed systems.Comment: This is a contribution to the Vadim Kuznetsov Memorial Issue on Integrable Systems and Related Topics, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Continuous Symmetries of the Lattice Potential KdV Equation

In this paper we present a set of results on the integration and on the symmetries of the lattice potential Korteweg-de Vries (lpKdV) equation. Using its associated spectral problem we construct the soliton solutions and the Lax technique enables us to provide infinite sequences of generalized symmetries. Finally, using a discrete symmetry of the lpKdV equation, we construct a large class of non-autonomous generalized symmetries.Comment: 20 pages, submitted to Jour. Phys.

Algebraic extensions of Gaudin models

We perform a In\"on\"u--Wigner contraction on Gaudin models, showing how the integrability property is preserved by this algebraic procedure. Starting from Gaudin models we obtain new integrable chains, that we call Lagrange chains, associated to the same linear $r$-matrix structure. We give a general construction involving rational, trigonometric and elliptic solutions of the classical Yang-Baxter equation. Two particular examples are explicitly considered: the rational Lagrange chain and the trigonometric one. In both cases local variables of the models are the generators of the direct sum of $N$ $\mathfrak{e}(3)$ interacting tops.Comment: 15 pages, corrected formula