Solutions of Adler's lattice equation associated with 2-cycles of the Backlund transformation

Details Solutions of Adler's lattice equation associated with 2-cycles of the Backlund transformation Solutions of Adler's lattice equation associated with 2-cycles of the Backlund transformation

2007-10-14

arxiv.org/abs/0710.2643v1

Nonlinear Sciences

Exactly Solvable and Integrable Systems

10 pages, contribution to the NEEDS 2007 proceedings

Scientific paper

The BT of Adler's lattice equation is inherent in the equation itself by virtue of its multidimensional consistency. We refer to a solution of the equation that is related to itself by the composition of two BTs (with different Backlund parameters) as a 2-cycle of the BT. In this article we will show that such solutions are associated with a commuting one-parameter family of rank-2 (i.e., 2-variable), 2-valued mappings. We will construct the explicit solution of the mappings within this family and hence give the solutions of Adler's equation that are 2-cycles of the BT.

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