### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 073, 26 pages      arXiv:1703.09963      https://doi.org/10.3842/SIGMA.2017.073
Contribution to the Special Issue on Symmetries and Integrability of Difference Equations

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

Ismagil Habibullin ab and Mariya Poptsova a
a) Ufa Institute of Mathematics, 112 Chernyshevsky Str., Ufa 450008, Russia
b) Bashkir State University, 32 Validy Str., Ufa 450076, Russia

Received March 30, 2017, in final form August 24, 2017; Published online September 07, 2017

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.

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

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References

1. Adler V.E., Habibullin I.T., Boundary conditions for integrable chains, Funct. Anal. Appl. 31 (1997), 75-85.
2. Adler V.E., Shabat A.B., Yamilov R.I., Symmetry approach to the integrability problem, Theoret. and Math. Phys. 125 (2000), 1603-1661.
3. Bogdanov L.V., Konopelchenko B.G., Grassmannians ${\rm Gr}(N-1,N+1)$, closed differential $N-1$-forms and $N$-dimensional integrable systems, J. Phys. A: Math. Theor. 46 (2013), 085201, 17 pages, arXiv:1208.6129.
4. Ferapontov E.V., Laplace transforms of hydrodynamic-type systems in Riemann invariants, Theoret. and Math. Phys. 110 (1997), 68-77, solv-int/9705017.
5. Ferapontov E.V., Khusnutdinova K.R., On the integrability of $(2+1)$-dimensional quasilinear systems, Comm. Math. Phys. 248 (2004), 187-206, nlin.SI/0305044.
6. Ferapontov E.V., Khusnutdinova K.R., Tsarev S.P., On a class of three-dimensional integrable Lagrangians, Comm. Math. Phys. 261 (2006), 225-243, nlin.SI/0407035.
7. Gubbiotti G., Scimiterna C., Yamilov R.I., Darboux integrability of trapezoidal $H^4$ and $H^6$ families of lattice equations II: general solutions, arXiv:1704.05805.
8. Gürel B., Habibullin I.T., Boundary conditions for two-dimensional integrable chains, Phys. Lett. A 233 (1997), 68-72.
9. Habibullin I.T., Characteristic Lie rings, finitely-generated modules and integrability conditions for $(2+1)$-dimensional lattices, Phys. Scripta 87 (2013), 065005, 5 pages, arXiv:1208.5302.
10. Habibullin I.T., Pekcan A., Characteristic Lie algebra and classification of semidiscrete models, Theoret. and Math. Phys. 151 (2007), 781-790, nlin.SI/0610074.
11. Levi D., Winternitz P., Lie point symmetries and commuting flows for equations on lattices, J. Phys. A: Math. Gen. 35 (2002), 2249-2262, math-ph/0112007.
12. Ma\ nas M., Martínez Alonso L., Álvarez-Fernández C., The multicomponent 2D Toda hierarchy: discrete flows and string equations, Inverse Problems 25 (2009), 065007, 31 pages, arXiv:0809.2720.
13. Mikhailov A.V., Shabat A.B., Sokolov V.V., The symmetry approach to classification of integrable equations, in What is Integrability?, Springer Ser. Nonlinear Dynam., Springer, Berlin, 1991, 115-184.
14. Mikhailov A.V., Yamilov R.I., Towards classification of $(2+1)$-dimensional integrable equations. Integrability conditions. I, J. Phys. A: Math. Gen. 31 (1998), 6707-6715.
15. Moser J., Finitely many mass points on the line under the influence of an exponential potential-an integrable system, in Dynamical Systems, Theory and Applications (Rencontres, Battelle Res. Inst., Seattle, Wash., 1974), Lecture Notes in Phys., Vol. 38, Springer, Berlin, 1975, 467-497.
16. Odesskii A.V., Sokolov V.V., Integrable $(2+1)$-dimensional systems of hydrodynamic type, Theoret. and Math. Phys. 163 (2010), 549-586, arXiv:1009.2778.
17. Pavlov M.V., Popowicz Z., On integrability of a special class of two-component $(2+1)$-dimensional hydrodynamic-type systems, SIGMA 5 (2009), 011, 10 pages, arXiv:0901.4312.
18. Pogrebkov A.K., Commutator identities on associative algebras and the integrability of nonlinear evolution equations, Theoret. and Math. Phys. 154 (2008), 405-417, nlin.SI/0703018.
19. Shabat A.B., Higher symmetries of two-dimensional lattices, Phys. Lett. A 200 (1995), 121-133.
20. Shabat A.B., Yamilov R.I., To a transformation theory of two-dimensional integrable systems, Phys. Lett. A 227 (1997), 15-23.
21. Smirnov S.V., Semidiscrete Toda lattices, Theoret. and Math. Phys. 172 (2012), 1217-1231, arXiv:1203.1764.
22. Smirnov S.V., Darboux integrability of discrete two-dimensional Toda lattices, Theoret. and Math. Phys. 182 (2015), 189-210, arXiv:1410.0319.
23. Yamilov R., Symmetries as integrability criteria for differential difference equations, J. Phys. A: Math. Gen. 39 (2006), R541-R623.
24. Zakharov V.E., Manakov S.V., Construction of higher-dimensional nonlinear integrable systems and of their solutions, Funct. Anal. Appl. 19 (1985), 89-101.
25. Zheltukhin K., Zheltukhina N., Semi-discrete hyperbolic equations admitting five dimensional characteristic $x$-ring, J. Nonlinear Math. Phys. 23 (2016), 351-367, arXiv:1604.00221.
26. Zheltukhin K., Zheltukhina N., Bilen E., On a class of Darboux-integrable semidiscrete equations, Adv. Difference Equ. (2017), 182, 14 pages.
27. Zhiber A.V., Murtazina R.D., Habibullin I.T., Shabat A.B., Characteristic Lie rings and integrable models in mathematical physics, Ufa Math. J. 4 (2012), 89-101.
28. Zhiber A.V., Murtazina R.D., Habibullin I.T., Shabat A.B., Characteristic Lie rings and nonlinear integrable equations, Institute of Computer Science, Moscow - Izhevsk, 2012.
29. Zhiber A.V., Sokolov V.V., Exactly integrable hyperbolic equations of Liouville type, Russ. Math. Surv. 56 (2001), no. 1, 61-101.