Classification of a Subclass of TwoDimensional Lattices via Characteristic Lie Rings
Abstract
The main goal of the article is testing a new classification algorithm. To this end we apply it to a relevant problem of describing the integrable cases of a subclass of twodimensional lattices. By imposing the cutoff conditions u_{1}=c_0 and u_{N+1}=c_1 we reduce the lattice u_{n,xy}=α(u_{n+1},u_n,u_{n1})u_{n,x}u_{n,y} to a finite system of hyperbolic type PDE. Assuming that for each natural N the obtained system is integrable in the sense of Darboux we look for α. To detect the Darboux integrability of the hyperbolic type system we use an algebraic criterion of Darboux integrability which claims that the characteristic Lie rings of such a system must be of finite dimension. We prove that up to the point transformations only one lattice in the studied class passes the test. The lattice coincides with the earlier found FerapontovShabatYamilov equation. The onedimensional reduction x=y of this lattice passes also the symmetry integrability test.
 Publication:

SIGMA
 Pub Date:
 September 2017
 DOI:
 10.3842/SIGMA.2017.073
 arXiv:
 arXiv:1703.09963
 Bibcode:
 2017SIGMA..13..073H
 Keywords:

 twodimensional integrable lattice;
 cutoff boundary condition;
 open chain;
 Darboux integrable system;
 characteristic Lie ring;
 Nonlinear Sciences  Exactly Solvable and Integrable Systems;
 37K10;
 37K30;
 37D99
 EPrint:
 SIGMA 13 (2017), 073, 26 pages