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TMF, 1982, Volume 51, Number 1, Pages 10–21 (Mi tmf2380)  

This article is cited in 49 scientific papers (total in 49 papers)

The group of internal symmetries and the conditions of integrability of two-dimensional dynamical systems

A. N. Leznov, V. G. Smirnov, A. B. Shabat


Abstract: The concept of the characteristic algebra of a system of equations of the form $u_{z\overline{z}}=F(u)$ is introduced. This algebra is associated with Lie–Bäcklund transformations. The conditions of integrability of such systems are formulated. It is shown that the case of integrability in quadrature corresponds to finite dimensionality of the characteristic algebra, while the case of integrability by the inverse scattering technique corresponds to this algebra's having a finite-dimensional representation. These requirements determine the form of the right-hand side $F$ for integrable systems.

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English version:
Theoretical and Mathematical Physics, 1982, 51:1, 322–330

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Received: 23.01.1981

Citation: A. N. Leznov, V. G. Smirnov, A. B. Shabat, “The group of internal symmetries and the conditions of integrability of two-dimensional dynamical systems”, TMF, 51:1 (1982), 10–21; Theoret. and Math. Phys., 51:1 (1982), 322–330

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. N. Leznov, M. V. Saveliev, “Two-dimensional nonlinear equations of the string type and their complete integration”, Theoret. and Math. Phys., 54:3 (1983), 209–218  mathnet  crossref  mathscinet  zmath  isi
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    6. A. N. Leznov, V. I. Man'ko, S. M. Chumakov, “Symmetries and soliton solutions of nonlinear equations”, Theoret. and Math. Phys., 63:1 (1985), 356–365  mathnet  crossref  mathscinet  zmath  isi
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    10. A. N. Leznov, M. A. Mukhtarov, “Internal symmetry algebra of exactly integrable dynamical systems in the quantum domain”, Theoret. and Math. Phys., 71:1 (1987), 370–375  mathnet  crossref  mathscinet  isi
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    26. O. S. Kostrigina, “Dvukhkomponentnye giperbolicheskie sistemy uravnenii eksponentsialnogo tipa s konechnomernoi kharakteristicheskoi algebroi Li”, Ufimsk. matem. zhurn., 1:3 (2009), 57–64  mathnet  zmath
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    28. V. de Alfaro, A. T. Filippov, “Multiexponential models of $(1+1)$-dimensional dilaton gravity and Toda–Liouville integrable models”, Theoret. and Math. Phys., 162:1 (2010), 34–56  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
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    30. Anatolii V. Zhiber, Olga S. Kostrigina, “Kharakteristicheskie algebry nelineinykh giperbolicheskikh sistem uravnenii”, Zhurn. SFU. Ser. Matem. i fiz., 3:2 (2010), 173–184  mathnet
    31. Habibullin I.T., Gudkova E.V., “Classification of integrable discrete Klein-Gordon models”, Phys Scripta, 83:4 (2011), 045003  crossref  isi
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    37. A. V. Zhiber, R. D. Murtazina, I. T. Khabibullin, A. B. Shabat, “Kharakteristicheskie koltsa Li i integriruemye modeli matematicheskoi fiziki”, Ufimsk. matem. zhurn., 4:3 (2012), 17–85  mathnet  mathscinet
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    39. Kuznetsova M.N., “Preobrazovanie laplasa i nelineinye giperbolicheskie sistemy”, Vestnik bashkirskogo universiteta, 17:4 (2012), 1653–1657  elib
    40. Habibullin I., “Characteristic Lie Rings, Finitely-Generated Modules and Integrability Conditions for (2+1)-Dimensional Lattices”, Phys. Scr., 87:6 (2013), 065005  crossref  isi
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    42. S. V. Smirnov, “Darboux integrability of discrete two-dimensional Toda lattices”, Theoret. and Math. Phys., 182:2 (2015), 189–210  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    43. Demskoi D.K. Tran D.T., “Darboux integrability of determinant and equations for principal minors”, Nonlinearity, 29:7 (2016), 1973–1991  crossref  mathscinet  zmath  isi  elib  scopus
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    47. Zheltukhin K. Zheltukhina N. Bilen E., “On a Class of Darboux-Integrable Semidiscrete Equations”, Adv. Differ. Equ., 2017, 182  crossref  isi
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    49. S. Ya. Startsev, “Structure of set of symmetries for hyperbolic systems of Liouville type and generalized Laplace invariants”, Ufa Math. J., 10:4 (2018), 103–110  mathnet  crossref  isi
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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