2000 character limit reached
On integrable reductions of two-dimensional Toda-type lattices
Published 17 May 2024 in nlin.SI | (2405.10666v1)
Abstract: The article considers lattices of the two-dimensional Toda type, which can be interpreted as dressing chains for spatially two-dimensional generalizations of equations of the class of nonlinear Schr\"odinger equations. The well-known example of this kind of generalization is the Davey-Stewartson equation. It turns out that the finite-field reductions of these lattices, obtained by imposing cutoff boundary conditions of an appropriate type, are Darboux integrable, i.e., they have complete sets of characteristic integrals. An algorithm for constructing complete sets of characteristic integrals of finite field systems using Lax pairs and Miura-type transformations is discussed.
- A.V.Mikhailov, R.I.Yamilov, “On integrable two-dimensional generalizations of nonlinear Schrödinger type equations”, Physics Letters, Section A: General, Atomic and Solid State Physics, 230:5–6 (1997), 295–300.
- A.B.Shabat, R.I.Yamilov, “To a transformation theory of two-dimensional integrable systems”, Phys. Lett., A, 227:1–2 (1997), 15–23.
- K.Ueno and K.Takasaki, Adv. Stud. Pure Math. 4 (1984) 1.
- Laplace P S 1776 Recherches sur le calcul intégral aux différences partielles (Mémoires de l’académie royale de Sciences de Paris).
- V.Volterra, “Lecons sur la theorie Mathematique de la Lutte pour la Vie”. Paris: Gautheier-Villars, 1931.
- B.N.Breizman, V.E.Zakharov, S.L.Musher, “On kinetics of stimulated scattering of Langmuir waves by plasma ions”- Zhurnal Ehksperimental’noj i Teoreticheskoj Fiziki 64 (4), 1297-1313, 1973.
- V.E.Zakharov, S.L.Musher, A.M.Rubenchik, “Nonlinear stage of parametric wave excitation in a plasma”.- Jetp Lett 19 (5), 151-152, 1974.
- I.T.Habibullin, “Characteristic Lie rings, finitely-generated modules and integrability conditions for (2+ 1)-dimensional lattices”, Physica Scripta, 87:6 (2013), 065005, IOP Publishing.
- M.N.Poptsova, I.T.Habibullin, “Algebraic properties of quasilinear two-dimensional lattices connected with integrability”, Ufa Math. J., 10:3 (2018), 86–105.
- A.V.Zhiber, R.D.Murtazina, I.T.Habibullin, A.B.Shabat, “Characteristic Lie rings and nonlinear integrable equations”, M.Izhevsk, 2012).
- A.B.Shabat, R.I.Yamilov. “Exponential systems of type I and Cartan matrices” // Preprint. Bashkir Branch of Academy of Sciences of USSR. (1981).
- 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.
- M.N.Kuznetsova, “Classification of a subclass of quasilinear two-dimensional lattices by means of characteristic algebras”, Ufa Math. J., 11:3 (2019), 109–131.
- E. V. Ferapontov, I. T. Habibullin, M. N. Kuznetsova and V. S. Novikov, “On a class of 2D integrable lattice equations”. J. Math. Phys. 61, 073505 (2020); doi: 10.1063/5.0013697.
- M.N.Kuznetsova, “Lax Pair for a Novel Two-Dimensional Lattice”, SIGMA, 17 (2021), 088, 13 pp.
- I.T.Habibullin, M.N.Poptsova, “Classification of a Subclass of Two-Dimensional Lattices via Characteristic Lie Rings”, SIGMA, 13 (2017), 73 , 26 pp.
- S.V.Smirnov, “Darboux integrability of discrete two-dimensional Toda lattices”, Theoret. and Math. Phys., 182:2 (2015), 189–210.
- M.Gurses, A.V.Zhiber, I.T.Habibullin, “Characteristic Lie rings of differential equations”, Ufimsk. Mat. Zh., 4:1 (2012), 53–62.
- D.K.Demskoi, “Integrals of open two-dimensional lattices”, Theoret. and Math. Phys., 163:1 (2010), 466–471.
- I.M.Anderson and N.Kamran 1997 “The variational bicomplex for second order scalar partial differential equations in the plane” Duke Math. J. 87 265-319.
- A.V.Zhiber and V.V.Sokolov 2001 “Exactly integrable hyperbolic equations of Liouville type” Russ. Math. Surv. 56 61-101.
- A.M.Gurieva, A.V.Zhiber, “Laplace Invariants of Two-Dimensional Open Toda Lattices”, Theoret. and Math. Phys., 138:3 (2004), 338–355.
- I.T.Habibullin, A.R.Khakimova, “Characteristic Lie algebras of integrable differential-difference equations in 3D”, Journal of Physics A: Mathematical and Theoretical, 54:29 (2021), 295202 , 34 pp.
- M. N. Kuznetsova, “Construction of localized particular solutions of chains with three independent variables”, Theoret. and Math. Phys., 216:2 (2023), 1158–1167.
- M. N. Kuznetsova, I. T. Habibullin, A. R. Khakimova, “On the problem of classifying integrable chains with three independent variables”, Theoret. and Math. Phys., 215:2 (2023), 667–690.
- A.V. Mikhailov, “Integrability of a Two-Dimensional Generalization of the Toda Chain”. Jetp Lett., 30:7 (1979), 414–418.
- E.V.Ferapontov, “Laplace transformations of hydrodynamic-type systems in Riemann invariants”, Theoret. and Math. Phys., 110:1 (1997), 68–77.
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