An 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-Backlund transformations for such lattice, and the corresponding formulas describing their superposition. We finally use these Darboux-Backlund 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.

arXiv comment14 pages, 4 figures, to appear in Phys. Rev. E http://pre.aps.org/

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• Integrable dynamics of Toda-type on the square and triangular lattices
• Darboux transformations for linear operators on two dimensional regular lattices
• Solitons of the (2+2)-dimensional Toda lattice
• Linear problems and Bäcklund transformations for the Hirota-Ohta system
• Darboux transformations for a 6-point scheme