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Classification of a subclass of quasilinear twodimensional 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 twodimensional lattices $u_{n,xy} = f(u_{n+1},u_n,u_{n1}, u_{n,x},u_{n,y})$. 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 semiinfinite 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 LieRinehart 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 $$u_{n,xy}=p(u_{n1},u_n,u_{n+1}) u_{n,x} + r(u_{n1},u_n,u_{n+1})u_{n,y} +q(u_{n1},u_n,u_{n+1}).$$
Keywords:
twodimensional lattice, integrable reduction, characteristic Lie algebra, degenerate cutting off condition, Darboux integrable system, $x$integral.
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Ufa Mathematical Journal, 2019, 11:3, 109–131 (PDF, 492 kB); https://doi.org/10.13108/2019113109
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UDC:
517.9
MSC: 37K10, 37K30, 37D99 Received: 03.04.2019
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Citation:
M. N. Kuznetsova, “Classification of a subclass of quasilinear twodimensional 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
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This publication is cited in the following articles:

I. T. Habibullin, M. N. Kuznetsova, “A classification algorithm for integrable twodimensional lattices
via Lie–Rinehart algebras”, Theoret. and Math. Phys., 203:1 (2020), 569–581

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