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 SIGMA, 2017, Volume 13, 073, 26 pages (Mi sigma1273)

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

Ismagil Habibullin, Mariya Poptsova

Ufa Institute of Mathematics, 112 Chernyshevsky Str., Ufa 450008, Russia

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.

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

DOI: https://doi.org/10.3842/SIGMA.2017.073

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MSC: 37K10; 37K30; 37D99
Received: March 30, 2017; in final form August 24, 2017; Published online September 7, 2017
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Citation: Ismagil Habibullin, Mariya Poptsova, “Classification of a Subclass of Two-Dimensional Lattices via Characteristic Lie Rings”, SIGMA, 13 (2017), 073, 26 pp.

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\Bibitem{HabKuz17} \by Ismagil~Habibullin, Mariya~Poptsova \paper Classification of a Subclass of Two-Dimensional Lattices via Characteristic Lie Rings \jour SIGMA \yr 2017 \vol 13 \papernumber 073 \totalpages 26 \mathnet{http://mi.mathnet.ru/sigma1273} \crossref{https://doi.org/10.3842/SIGMA.2017.073} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000410663200001} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85029173515} 

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This publication is cited in the following articles:
1. M. N. Poptsova, I. T. Habibullin, “Algebraic properties of quasilinear two-dimensional lattices connected with integrability”, Ufa Math. J., 10:3 (2018), 86–105
2. M. N. Poptsova, “Simmetrii odnoi periodicheskoi tsepochki”, Kompleksnyi analiz. Matematicheskaya fizika, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 162, VINITI RAN, M., 2019, 80–84
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