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 Ufimsk. Mat. Zh., 2018, Volume 10, Issue 3, Pages 89–109 (Mi ufa441)

Algebraic properties of quasilinear two-dimensional lattices connected with integrability

M. N. Poptsovaa, I. T. Habibullinba

a Institute of Mathematics, Ufa Federal Research Center, RAS, Chernyshevsky str. 112, 450008, Ufa, Russia
b Bashkir State University, Validy str. 32, 450077, Ufa, Russia

Abstract: In the paper we discuss a classification method for nonlinear integrable equations with three independent variables based on the notion of the integrable reductions. We call an equation integrable if it admits a large class of reductions being Darboux integrable systems of hyperbolic type equations with two independent variables. The most natural and convenient object to be studied in the framework of this scheme is the class of two dimensional lattices generalizing the well-known Toda lattice. In the present article we study the quasilinear lattices of the form
\begin{align*} u_{n,xy}=&\alpha(u_{n+1} ,u_n,u_{n-1} )u_{n,x}u_{n,y} + \beta(u_{n+1},u_n,u_{n-1})u_{n,x}
&+\gamma(u_{n+1} ,u_n,u_{n-1} )u_{n,y}+\delta(u_{n+1} ,u_n,u_{n-1}). \end{align*}
We specify the coefficients of the lattice assuming that there exist cutting off conditions which reduce the lattice to a Darboux integrable hyperbolic type system of the arbitrarily high order. Under some extra assumption of nondegeneracy we describe the class of the lattices integrable in the above sense. There are new examples in the obtained list of chains.

Keywords: two-dimensional integrable lattice, $x$-integral, integrable reduction, cut-off condition, open chain, Darboux integrable system, characteristic Lie algebra.

 Funding Agency Grant Number Russian Science Foundation 15-11-20007 The authors gratefully acknowledge financial support from a Russian Science Foundation grant (project 15-11-20007).

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English version:
Ufa Mathematical Journal, 2018, 10:3, 86–105 (PDF, 461 kB); https://doi.org/10.13108/2018-10-3-86

Bibliographic databases:

UDC: 517.9
MSC: 37K10, 37K30, 37D99

Citation: M. N. Poptsova, I. T. Habibullin, “Algebraic properties of quasilinear two-dimensional lattices connected with integrability”, Ufimsk. Mat. Zh., 10:3 (2018), 89–109; Ufa Math. J., 10:3 (2018), 86–105

Citation in format AMSBIB
\Bibitem{KuzHab18} \by M.~N.~Poptsova, I.~T.~Habibullin \paper Algebraic properties of quasilinear two-dimensional lattices connected with integrability \jour Ufimsk. Mat. Zh. \yr 2018 \vol 10 \issue 3 \pages 89--109 \mathnet{http://mi.mathnet.ru/ufa441} \transl \jour Ufa Math. J. \yr 2018 \vol 10 \issue 3 \pages 86--105 \crossref{https://doi.org/10.13108/2018-10-3-86} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000457365400007} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85057011860} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. 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
2. Ufa Math. J., 11:3 (2019), 109–131
3. I. T. Habibullin, M. N. Kuznetsova, “A classification algorithm for integrable two-dimensional lattices via Lie–Rinehart algebras”, Theoret. and Math. Phys., 203:1 (2020), 569–581
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