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An integrable generalization of the Toda law to the square lattice

P. M. Santini·M. Nieszporski·A. Doliwa
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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.

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P. M. Santini, M. Nieszporski, A. Doliwa. 2004-09-24. An integrable generalization of the Toda law to the square lattice. https://doi.org/10.1103/physreve.70.056615

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