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

M. N. Kuznetsova

Ufa mathematical journal, Tome 11 (2019) no. 3, pp. 109-131

Résumé

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 $-\infty$ and $0$, 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 $$u_{n,xy}=p(u_{n-1},u_n,u_{n+1}) u_{n,x} + r(u_{n-1},u_n,u_{n+1})u_{n,y} +q(u_{n-1},u_n,u_{n+1}).$$

Détail

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

Bibliographie

[1] E. V. Ferapontov, K. R. Khusnutdinova, “On the integrability of (2+1)-dimensional quasilinear systems”, Comm. Math. Phys., 248:1 (2004), 187–206 | DOI | MR | Zbl

[2] E. V. Ferapontov, K. R. Khusnutdinova, M. V. Pavlov, “Classification of integrable (2+1)-dimensional quasilinear hierarchies”, Theor. Math. Phys., 144:1 (2005), 907–915 | DOI | MR | Zbl

[3] E. V. Ferapontov, K. R. Khusnutdinova, S. P. Tsarev, “On a class of three-dimensional integrable Lagrangians”, Comm. Math. Phys., 261:1 (2006), 225–243 | DOI | MR | Zbl

[4] A. V. Odesskii, V. V. Sokolov, “Integrable (2+1)-dimensional systems of hydrodynamic type”, Theor. Math. Phys., 163:2 (2010), 549–586 | DOI | Zbl

[5] L. V. Bogdanov, B. G. Konopelchenko, “Grassmannians $Gr(N-1,N+1)$, closed differential $N-1$-forms and $N$-dimensional integrable systems”, J. Phys. A: Math. Theor., 46:8 (2013), 085201 | DOI | MR | Zbl

[6] M. V. Pavlov, Z. Popowicz, “On integrability of a special class of two-component (2+1)-dimensional hydrodynamic-type systems”, SIGMA, 5 (2009), 011 | MR | Zbl

[7] A. K. Pogrebkov, “Commutator identities on associative algebras and the integrability of nonlinear evolution equations”, Theor. Math. Phys., 154:3 (2008), 405–417 | DOI | MR | Zbl

[8] M. Mañas, L. M. Alonso, C. Alvarez-Fernández, “The multicomponent 2D Toda hierarchy: discrete flows and string equations”, Inverse Problems, 25 (2009), 065007 | DOI | MR | Zbl

[9] V. E. Zakharov, S. V. Manakov, “Construction of higher-dimensional nonlinear integrable systems and of their solutions”, Funct. Anal. Appl., 19:2 (1985), 89–101 | DOI | MR | Zbl

[10] I. S. Krasil'shchik, A. Sergyeyev, O. I. Morozov, “Infinitely many nonlocal conservation laws for the $ABC$ equation with $A+B+C\ne0$”, Calc. Var. PDEs, 55:5 (2016), 123 | DOI | MR | Zbl

[11] A. B. Shabat, R. I. Yamilov, Exponential systems of type I and the Cartan matrix, Preprint, Bashkir branch of AS USSR, Ufa, 1981 (in Russian)

[12] A. N. Leznov, V. G. Smirnov, A. B. Shabat, “The group of internal symmetries and the conditions of integrability of two-dimensional dynamical systems”, Theor. Math. Phys., 51:1 (1982), 322–330 | DOI | MR | Zbl

[13] F. H. Mukminov, A. V. Zhiber, “Quadratic systems, symmetries, characteristic and complete algebras”, Problems of mathematical physics and the asymptotics of their solutions, Bashkir Scientific Center of the Ural Branch AS USSR, Ufa, 1991, 14–32 (in Russian)

[14] A. V. Zhiber, R. D. Murtazina, I. T. Habibullin, A. B. Shabat, Characteristic Lie rings and nonlinear integrable equations, Inst. Comp. Stud., M., 2012 (in Russian) | MR

[15] A. V. Zhiber, R. D. Murtazina, I. T. Habibullin, A. B. Shabat, “Characteristic Lie rings and integrable models in mathematical physics”, Ufa Math. J., 4:3 (2012), 17–85 | MR | Zbl

[16] I. T. Habibullin, “Characteristic Lie rings, finitely-generated modules and integrability conditions for (2+1)-dimensional lattices”, Physica Scripta, 87:6 (2013), 065005 | DOI | Zbl

[17] I. T. Habibullin, M. N. Poptsova (Kuznetsova), “Classification of a subclass of two-dimensional lattices via characteristic lie rings”, SIGMA, 13 (2017), 073 | MR | Zbl

[18] I. T. Habibullin, M. N. Poptsova (Kuznetsova), “Algebraic properties of quasilinear two-dimensional lattices connected with integrability”, Ufa Math. J., 10:3 (2018), 86–105 | DOI | MR

[19] A. B. Shabat, “Higher symmetries of two-dimensional lattices”, Phys. Lett. A, 200:2 (1995), 121–133 | DOI | MR | Zbl

[20] G. Rinehart, “Differential forms for general commutative algebras”, Trans. Amer.Math. Soc., 108:2 (1963), 195–222 | DOI | MR | Zbl

[21] D. Millionshchikov, “Lie Algebras of Slow Growth and Klein-Gordon PDE”, Algebras Represent. Theor., 21:5 (2018), 1037–1069 | DOI | MR | Zbl

[22] A. B. Shabat, R. I. Yamilov, “To a transformation theory of two-dimensional integrable systems”, Phys. Lett. A, 227:1–2 (1997), 15–23 | DOI | MR | Zbl

[23] R. Yamilov, “Symmetries as integrability criteria for differential difference equations”, J. Phys. A: Math. Gen., 39:45 (2006), 541–623 | DOI | MR

[24] I. T. Habibullin, A. Pekcan, “Characteristic Lie algebra and classification of semidiscrete models”, Theor. Math. Phys., 151:3 (2007), 781–790 | DOI | MR | Zbl

[25] A. N. Leznov, V. G. Smirnov, A. B. Shabat, “The group of internal symmetries and the conditions of integrability of two-dimensional dynamical systems”, Theor. Math. Phys., 51:1 (1982), 322–330 | DOI | MR | Zbl