An integrable generalization of the Toda law to the square lattice

Details An integrable generalization of the Toda law to the square lattice An integrable generalization of the Toda law to the square lattice

2004-09-24

arxiv.org/abs/nlin/0409050v1

Nonlinear Sciences

Exactly Solvable and Integrable Systems

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

Scientific paper

10.1103/PhysRevE.70.056615

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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