Recent results from the Pierre Auger Observatory

In this paper some recent results from the Pierre Auger Collaboration are presented. These are the measurement of the energy spectrum of cosmic rays over a wide range of energies ($10^{17.5}$ to above $10^{20}$ eV), studies of the cosmic-ray mass composition with the fluorescence and surface detector of the Observatory, the observation of a large-scale anisotropy in the arrival direction of cosmic rays above 8 x $10^{18}$ eV and indications of anisotropy at intermediate angular scales above 4 x $10^{19}$ eV. The astrophysical implications of the spectrum and composition results are also discussed. Finally the progress of the upgrade of the Observatory, AugerPrime is presented.Comment: 20th International Symposium on Very High Energy Cosmic Ray Interactions (ISVHECRI 2018

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.

Spherical geometry and integrable systems

We prove that the cosine law for spherical triangles and spherical tetrahedra defines integrable systems, both in the sense of multidimensional consistency and in the sense of dynamical systems.Comment: 15 pages, 5 figure