V. E. Adler, S. Ya. Startsev, “Discrete analogues of the Liouville equation”, TMF, 121:2 (1999), 271–284; Theoret. and Math. Phys., 121:2 (1999), 1484

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Teoreticheskaya i Matematicheskaya Fizika, 1999, Volume 121, Number 2, Pages 271–284
DOI: https://doi.org/10.4213/tmf808 (Mi tmf808)

This article is cited in 81 scientific papers (total in 81 papers)

Discrete analogues of the Liouville equation

V. E. Adler, S. Ya. Startsev

Institute of Mathematics with Computing Centre, Ufa Science Centre, Russian Academy of Sciences

Full-text PDF (241 kB) Citations (81)

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DOI: https://doi.org/10.4213/tmf808

Abstract: The notion of Laplace invariants is generalized to lattices and discrete equations that are difference analogues of hyperbolic partial differential equations with two independent variables. The sequence of Laplace invariants satisfies the discrete analogue of the two-dimensional Toda lattice. We prove that terminating this sequence by zeros is a necessary condition for the existence of integrals of the equation under consideration. We present formulas for the higher symmetries of equations possessing such integrals. We give examples of difference analogues of the Liouville equation.

Received: 16.02.1999

English version:
Theoretical and Mathematical Physics, 1999, Volume 121, Issue 2, Pages 1484–1495
DOI: https://doi.org/10.1007/BF02557219

Bibliographic databases:

Language: Russian

Citation: V. E. Adler, S. Ya. Startsev, “Discrete analogues of the Liouville equation”, TMF, 121:2 (1999), 271–284; Theoret. and Math. Phys., 121:2 (1999), 1484–1495

Citation in format AMSBIB

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Linking options:

https://www.mathnet.ru/eng/tmf808

https://doi.org/10.4213/tmf808

https://www.mathnet.ru/eng/tmf/v121/i2/p271

This publication is cited in the following 81 articles:

R. N. Garifullin, “Classification of semidiscrete equations of hyperbolic type. The case of fifth-order symmetries”, Theoret. and Math. Phys., 222:1 (2025), 10–19
S Ya Startsev, “Darboux integrability of hyperbolic partial differential equations: is it a property of integrals rather than equations?”, J. Phys. A: Math. Theor., 58:2 (2025), 025206
Xiaoxue Xu, Decong Yi, Liyuan Ma, “A novel solution to the generalized lattice Liouville equation”, Applied Mathematics Letters, 2024, 109115
I T Habibullin, K I Faizulina, A R Khakimova, “Laplace transformations and sine-Gordon type integrable PDE”, J. Phys. A: Math. Theor., 57:1 (2024), 015203
Giorgio Gubbiotti, “Algebraic entropy for systems of quad equations”, Open Communications in Nonlinear Mathematical Physics, Special Issue in Memory of... (2024)
Giorgio Gubbiotti, Andrew P Kels, Claude-M Viallet, “Algebraic entropy for hex systems”, Nonlinearity, 37:12 (2024), 125007
Hong-juan Tian, A. Silem, “Cauchy matrix approach to novel extended semidiscrete KP-type systems”, Theoret. and Math. Phys., 221:2 (2024), 1929–1939
A R Khakimova, K I Faizulina, “Reduction of the Laplace sequence and sine-Gordon type equations”, J. Phys. A: Math. Theor., 57:49 (2024), 495208
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
M. N. Kuznetsova, “Construction of localized particular solutions of chains with three independent variables”, Theoret. and Math. Phys., 216:2 (2023), 1158–1167
I. T. Habibullin, A. R. Khakimova, “On the classification of nonlinear integrable three-dimensional chains via characteristic Lie algebras”, Theoret. and Math. Phys., 217:1 (2023), 1541–1573
Ismagil T. Habibullin, Aigul R. Khakimova, Alfya U. Sakieva, “Miura-Type Transformations for Integrable Lattices in 3D”, Mathematics, 11:16 (2023), 3522
Sergey V Smirnov, “Integral preserving discretization of 2D Toda lattices”, J. Phys. A: Math. Theor., 56:26 (2023), 265204
Kostyantyn Zheltukhin, Natalya Zheltukhina, “On Construction of Darboux integrable discrete models”, Reports on Mathematical Physics, 92:3 (2023), 279
I. T. Habibullin, A. R. Khakimova, “Integrals and characteristic algebras for systems of discrete equations on a quadrilateral graph”, Theoret. and Math. Phys., 213:2 (2022), 1589–1612
I. T. Habibullin, A. R. Khakimova, “Algebraic reductions of discrete equations of Hirota-Miwa type”, Ufa Math. J., 14:4 (2022), 113–126
D. V. Millionshchikov, S. V. Smirnov, “Characteristic algebras and integrable exponential systems”, Ufa Math. J., 13:2 (2021), 41–69
Ufa Math. J., 13:2 (2021), 160–169
Ufa Math. J., 13:2 (2021), 170–186
Dmitry K. Demskoi, “The Lattice Sine-Gordon Equation as a Superposition Formula for an NLS-Type System”, SIGMA, 17 (2021), 108, 10 pp.

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

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