What is the relativistic Volterra lattice ?

We develop a systematic procedure of finding integrable ”relativistic” (regular one– parameter) deformations for integrable lattice systems. Our procedure is based on the integrable time discretizations and consists of three steps. First, for a given system one finds a local discretization living in the same hierarchy. Second, one considers this discretization as a particular Cauchy problem for a certain 2–dimensional lattice equation, and then looks for another meaningful Cauchy problems, which can be, in turn, interpreted as new discrete time systems. Third, one has to identify integrable hierarchies to which these new discrete time systems belong. These novel hierarchies are called then ”relativistic”, the small time step h playing the role of inverse speed of light. We apply this procedure to the Toda lattice (and recover the well–known relativistic Toda lattice), as well as to the Volterra lattice and a certain Bogoyavlensky lattice, for which the ”relativistic” deformations were not known previously. Centre for Complex Systems and Visualization, University of Bremen, Universitätsallee 29, 28359 Bremen, Germany; present address: Technische Universität Berlin, Fachbereich Mathematik, SFB 288, Sekr. MA 8–5, Str. des 17. Juni 136, 10623 Berlin, Germany; e-mail: suris @ sfb288.math.tu-berlin.de Dipartimento di Fisica, Universita di Roma Tre, Via Vasca Navale 84, 00146 Roma, Italy; e-mail: ragnisco @ amaldi.fis.uniroma3.it