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

SIGMA 17 (2021), 088, 13 pages      arXiv:2102.04207

Lax Pair for a Novel Two-Dimensional Lattice

Maria N. Kuznetsova
Institute of Mathematics, Ufa Federal Research Centre, Russian Academy of Sciences,112 Chernyshevsky Street, Ufa 450008, Russia

Received February 09, 2021, in final form September 15, 2021; Published online September 26, 2021

In paper by I.T. Habibullin and our joint paper the algorithm for classification of integrable equations with three independent variables was proposed. This method is based on the requirement of the existence of an infinite set of Darboux integrable reductions and on the notion of the characteristic Lie-Rinehart algebras. The method was applied for the classification of integrable cases of different subclasses of equations $u_{n,xy} = f(u_{n+1},u_n,u_{n-1}, u_{n,x},u_{n,y})$ of special forms. Under this approach the novel integrable chain was obtained. In present paper we construct Lax pair for the novel chain. To construct the Lax pair, we use the scheme suggested in papers by E.V. Ferapontov. We also study the periodic reduction of the chain.

Key words: Lax pair; two-dimensional lattice; integrable reduction; characteristic algebra; Lie-Rinehart algebra; Darboux integrable system; higher symmetry; $x$-integral.

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  1. Bogdanov L.V., Dunajski-Tod equation and reductions of the generalized dispersionless 2DTL hierarchy, Phys. Lett. A 376 (2012), 2894-2898, arXiv:1204.3780.
  2. Bogdanov L.V., Konopelchenko B.G., On dispersionless BKP hierarchy and its reductions, J. Nonlinear Math. Phys. 12 (2005), suppl. 1, 64-73, arXiv:nlin.SI/0411046.
  3. Calderbank D.M.J., Kruglikov B., Integrability via geometry: dispersionless differential equations in three and four dimensions, Comm. Math. Phys. 382 (2021), 1811-1841, arXiv:1612.02753.
  4. Demskoi D.K., Startsev S.Ya., On construction of symmetries from integrals of hyperbolic partial differential systems, J. Math. Phys. 136 (2006), 4378-4384.
  5. Doubrov B., Ferapontov E.V., Kruglikov B., Novikov V.S., On integrability in Grassmann geometries: integrable systems associated with fourfolds in ${\bf Gr}(3,5)$, Proc. Lond. Math. Soc. 116 (2018), 1269-1300, arXiv:1503.02274.
  6. Doubrov B., Ferapontov E.V., Kruglikov B., Novikov V.S., Integrable systems in four dimensions associated with six-folds in ${\rm Gr}(4,6)$, Int. Math. Res. Not. 2019 (2019), 6585-6613, arXiv:1705.06999.
  7. Ferapontov E.V., Laplace transforms of hydrodynamic-type systems in Riemann invariants, Theoret. and Math. Phys. 110 (1997), 68-77, arXiv:solv-int/9705017.
  8. Ferapontov E.V., Habibullin I.T., Kuznetsova M.N., Novikov V.S., On a class of 2D integrable lattice equations, J. Math. Phys. 61 (2020), 073505, 15 pages, arXiv:2005.06738.
  9. Ferapontov E.V., Hadjikos L., Khusnutdinova K.R., Integrable equations of the dispersionless Hirota type and hypersurfaces in the Lagrangian Grassmannian, Int. Math. Res. Not. 2010 (2010), 496-535, arXiv:0705.1774.
  10. Ferapontov E.V., Khusnutdinova K.R., Hydrodynamic reductions of multidimensional dispersionless PDEs: the test for integrability, J. Math. Phys. 45 (2004), 2365-2377, arXiv:nlin.SI/0312015.
  11. Ferapontov E.V., Kruglikov B.S., Dispersionless integrable systems in 3D and Einstein-Weyl geometry, J. Differential Geom. 97 (2014), 215-254, arXiv:1208.2728.
  12. Ferapontov E.V., Novikov V.S., Roustemoglou I., Towards the classification of integrable differential-difference equations in $2+1$ dimensions, J. Phys. A: Math. Theor. 46 (2013), 245207, 13 pages, arXiv:21303.3430.
  13. Habibullin I.T., Characteristic Lie rings, finitely-generated modules and integrability conditions for (2+1)-dimensional lattices, Phys. Scr. 87 (2013), 065005, 5 pages, arXiv:1208.5302.
  14. Habibullin I.T., Kuznetsova M.N., A classification algorithm for integrable two-dimensional lattices via Lie-Rinehart algebras, Theoret. and Math. Phys. 203 (2020), 569-581, arXiv:1907.12269.
  15. Habibullin I.T., Kuznetsova M.N., Sakieva A.U., Integrability conditions for two-dimensional Toda-like equations, J. Phys. A: Math. Theor. 53 (2020), 395203, 25 pages, arXiv:2005.09712.
  16. Habibullin I., Poptsova M., Classification of a subclass of two-dimensional lattices via characteristic Lie rings, SIGMA 13 (2017), 073, 26 pages, arXiv:1703.09963.
  17. Kiselev A.V., van de Leur J.W., Symmetry algebras of Lagrangian Liouville-type systems, Theoret. and Math. Phys. 162 (2010), 149-–162, arXiv:0902.3624.
  18. Kuznetsova M.N., Classification of a subclass of quasilinear two-dimensional lattices by means of characteristic algebras, Ufa Math. J. 11 (2019), 109-131.
  19. Leznov A.N., Smirnov V.G., Shabat A.B., The group of internal symmetries and the conditions of integrability of two-dimensional dynamical systems, Theoret. and Math. Phys. 51 (1982), 322-330.
  20. Millionshchikov D., Lie algebras of slow growth and Klein-Gordon PDE, Algebr. Represent. Theory 21 (2018), 1037-1069, arXiv:1711.03706.
  21. Pavlov M.V., Classifying integrable Egoroff hydrodynamic chains, Theoret. and Math. Phys. 138 (2004), 45-58.
  22. Rinehart G.S., Differential forms on general commutative algebras, Trans. Amer. Math. Soc. 108 (1963), 195-222.
  23. Shabat A.B., Yamilov R.I., To a transformation theory of two-dimensional integrable systems, Phys. Lett. A 227 (1997), 15-23.
  24. Startsev S.Ya., On differential substitutions of the Miura transformation type, Theoret. and Math. Phys. 116 (1998), 1001-1010.
  25. Startsev S.Ya., On the variational integrating matrix for hyperbolic systems, J. Math. Phys. 151 (2008), 3245-3253.
  26. Xenitidis P., Determining the symmetries of difference equations, Proc. R. Soc. Lond. A 474 (2018), 20180340, 20 pages.
  27. Zakharov V.E., Dispersionless limit of integrable systems in $2+1$ dimensions, in Singular Limits of Dispersive Waves (Lyon, 1991), NATO Adv. Sci. Inst. Ser. B: Phys., Vol. 320, Springer, Boston, MA, 1994, 165-174.
  28. Zhiber A.V., Kostrigina O.S., Exactly integrable models of wave processes, Vestnik USATU 9 (2007), no. 7, 83-89.
  29. 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.
  30. Zhiber A.V., Sokolov V.V., Startsev S.Ya., On nonlinear Darboux-integrable hyperbolic equations, Dokl. Akad. Nauk 343 (1995), 746-748.

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