title: Integrable generalization of the Toda law to the square lattice.
abstract: We generalize the Toda lattice (or Toda chain) equation to the square lattice, i.e., we construct an integrable nonlinear equation for a scalar field taking values on the square lattice and depending on a continuous (time) variable, characterized by an exponential law of interaction in both discrete directions of the square lattice. We construct the Darboux-Bäcklund transformations for such lattice, and the corresponding formulas describing their superposition. We finally use these Darboux-Bäcklund transformations to generate examples of explicit solutions of exponential and rational type. The exponential solutions describe the evolution of one and two smooth two-dimensional shock waves on the square lattice.
authors.affiliation: Dipartimento di Fisica, Università di Roma La Sapienza, and Istituto Nazionale di Fisica Nucleare, Sezione di Roma, Piazza le Aldo Moro 2, I-00185 Roma, Italy. paolo.santini@roma1.infn.it
authors.forename: P M
authors.initials: PM
authors.lastname: Santini
authors.forename: M
authors.initials: M
authors.lastname: Nieszporski
authors.forename: A
authors.initials: A
authors.lastname: Doliwa