Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 073, 26 pages      arXiv:1703.09963
Contribution to the Special Issue on Symmetries and Integrability of Difference Equations

Classification of a Subclass of Two-Dimensional Lattices via Characteristic Lie Rings

Ismagil Habibullin ab and Mariya Poptsova a
a) Ufa Institute of Mathematics, 112 Chernyshevsky Str., Ufa 450008, Russia
b) Bashkir State University, 32 Validy Str., Ufa 450076, Russia

Received March 30, 2017, in final form August 24, 2017; Published online September 07, 2017

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 two-dimensional lattices. By imposing the cut-off conditions $u_{-1}=c_0$ and $u_{N+1}=c_1$ we reduce the lattice $u_{n,xy}=\alpha(u_{n+1},u_n,u_{n-1})u_{n,x}u_{n,y}$ 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 $\alpha$. 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 Ferapontov-Shabat-Yamilov equation. The one-dimensional reduction $x=y$ of this lattice passes also the symmetry integrability test.

Key words: two-dimensional integrable lattice; cut-off boundary condition; open chain; Darboux integrable system; characteristic Lie ring.

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