Phys. Rev. E 70, 056615 (2004) [6 pages]Integrable generalization of the Toda law to the square latticeReceived 9 July 2004; published 23 November 2004 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. © 2004 The American Physical Society URL:
http://link.aps.org/doi/10.1103/PhysRevE.70.056615
DOI:
10.1103/PhysRevE.70.056615
PACS:
03.50.−z, 05.45.−a, 63.20.Pw, 05.50.+q
|
|
About | Terms and Conditions | Subscriptions | Search | Help
Use of the American Physical Society websites and journals implies that the user has read and agrees to our Terms and Conditions and any applicable Subscription Agreement. Physical Review®, Physical Review Letters®, Reviews of Modern Physics®, and Physical Review Special Topics® are trademarks of the American Physical Society.