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Integrable generalization of the Toda law to the square lattice
P M Santini1, M Nieszporski, A Doliwa
1Dipartimento 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
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.
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