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Ufimsk. Mat. Zh., 2019, Volume 11, Issue 3, Pages 110–131 (Mi ufa484)  

This article is cited in 1 scientific paper (total in 1 paper)

Classification of a subclass of quasilinear two-dimensional lattices by means of characteristic algebras

M. N. Kuznetsova

Institute of Mathematics, Ufa Federal Research Center, Russian Academy of Sciences, Chernyshevsky str. 112, 450008, Ufa, Russia

Abstract: We consider a classification problem of integrable cases of the Toda type two-dimensional lattices $u_{n,xy} = f(u_{n+1},u_n,u_{n-1}, u_{n,x},u_{n,y})$. The function $f = f(x_1,x_2,\cdots x_5)$ is assumed to be analytic in a domain $D\subset \mathbb{C}^5$. The sought function $u_n = u_n(x,y)$ depends on real $x$, $y$ and integer $n$. Equations with three independent variables are complicated objects for study and especially for classification. It is commonly accepted that for a given equation, the existence of a large class of integrable reductions indicates integrability. Our classification algorithm is based on this observation. We say that a constraint $u_0 = \varphi(x,y)$ defines a degenerate cutting off condition for the lattice if it divides this lattice into two independent semi-infinite lattices defined on the intervals $-\infty<n<0$ and $0<n<+\infty$, respectively. We call a lattice integrable if there exist cutting off boundary conditions allowing us to reduce the lattice to an infinite number of hyperbolic type systems integrable in the sense of Darboux. Namely, we require that lattice is reduced to a finite system of such kind by imposing degenerate cutting off conditions at two different points $n=N_1$, $n=N_2$ for arbitrary pair of integers $N_1$, $N_2$. Recall that a system of hyperbolic equations is called Darboux integrable if it admits a complete set of integrals in both characteristic directions. An effective criterion of the Darboux integrability of the system is connected with properties of an associated algebraic structures. More precisely, the characteristic Lie-Rinehart algebras assigned to both characteristic directions have to be of a finite dimension. Since the obtained hyperbolic system is of a very specific form, the characteristic algebras are effectively studied. Here we focus on a subclass of quasilinear lattices of the form
$$u_{n,xy}=p(u_{n-1},u_n,u_{n+1}) u_{n,x} + r(u_{n-1},u_n,u_{n+1})u_{n,y} +q(u_{n-1},u_n,u_{n+1}).$$

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

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English version:
Ufa Mathematical Journal, 2019, 11:3, 109–131 (PDF, 492 kB);

Bibliographic databases:

UDC: 517.9
MSC: 37K10, 37K30, 37D99
Received: 03.04.2019

Citation: M. N. Kuznetsova, “Classification of a subclass of quasilinear two-dimensional lattices by means of characteristic algebras”, Ufimsk. Mat. Zh., 11:3 (2019), 110–131; Ufa Math. J., 11:3 (2019), 109–131

Citation in format AMSBIB
\by M.~N.~Kuznetsova
\paper Classification of a subclass of quasilinear two-dimensional lattices by means of characteristic algebras
\jour Ufimsk. Mat. Zh.
\yr 2019
\vol 11
\issue 3
\pages 110--131
\jour Ufa Math. J.
\yr 2019
\vol 11
\issue 3
\pages 109--131

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    This publication is cited in the following articles:
    1. 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  mathnet  crossref  crossref  isi  elib
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