Classification of a Subclass of Two-Dimensional 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 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.

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Cited by in corpus (6)

• On a classification algorithm of the integrable two-dimensional lattices via Lie-Rinehart algebras
• Characteristic Lie Algebras of Integrable Differential-Difference Equations in 3D
• On a class of 2D integrable lattice equations
• Integral preserving discretization of 2D Toda lattices
• An algebraic criterion of the Darboux integrability of differential-difference equations and systems
• Lax Pair for a Novel Two-Dimensional Lattice