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Ufimsk. Mat. Zh., 2012, Volume 4, Issue 3, Pages 17–85 (Mi ufa156)  

This article is cited in 13 scientific papers (total in 13 papers)

Characteristic Lie rings and integrable models in mathematical physics

A. V. Zhibera, R. D. Murtazinab, I. T. Habibullina, A. B. Shabatc

a Institute of Mathematics with Computing Centre, Ufa Science Centre, Russian Academy of Sciences, Ufa, Russia
b Ufa State Aviation Technical University, Ufa, Russia
c L. D. Landau Institute for Theoretical Physics, Russian Academy of Sciences, Moscow, Russia

Abstract: Review is devoted to a systematic exposition of the algebraic approach to the study of nonlinear integrable partial differential equations and their discrete analogues, based on the concept of the characteristic vector field. A special attention is paid to the Darboux integrable equations. The problem of constructing higher symmetries of the equations, as well as their particular and general solutions is discussed. In particular, it is shown that the partial differential equation of hyperbolic type is integrated in quadratures if and only if its characteristic Lie rings in both directions are of finite dimension. For the hyperbolic type equations integrable by the inverse scattering method, the characteristic rings are of minimal growth. The possible applications of the concept of characteristic Lie rings to the systems of differential equations of hyperbolic type with more than two characteristic directions, to the equations of evolution type, and to ordinary differential equations are discussed.

Keywords: characteristic vector field, symmetry, Darboux integrability.

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Bibliographic databases:
UDC: 517.957
Received: 25.11.2011

Citation: A. V. Zhiber, R. D. Murtazina, I. T. Habibullin, A. B. Shabat, “Characteristic Lie rings and integrable models in mathematical physics”, Ufimsk. Mat. Zh., 4:3 (2012), 17–85

Citation in format AMSBIB
\by A.~V.~Zhiber, R.~D.~Murtazina, I.~T.~Habibullin, A.~B.~Shabat
\paper Characteristic Lie rings and integrable models in mathematical physics
\jour Ufimsk. Mat. Zh.
\yr 2012
\vol 4
\issue 3
\pages 17--85

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    This publication is cited in the following articles:
    1. Habibullin I., “Characteristic Lie Rings, Finitely-Generated Modules and Integrability Conditions for (2+1)-Dimensional Lattices”, Phys. Scr., 87:6 (2013), 065005  crossref  zmath  isi  elib  scopus
    2. V. M. Zhuravlev, “Matrichnye funktsionalnye podstanovki dlya integriruemykh dinamicheskikh sistem i uravneniya Landau–Lifshitsa”, Nelineinaya dinam., 10:1 (2014), 35–48  mathnet
    3. K. Zheltukhin, N. Zheltukhina, “On Existence of an X-Integral for a Semi-Discrete Chain of Hyperbolic Type”, XXIII International Conference on Integrable Systems and Quantum Symmetries ISQS-23, Journal of Physics Conference Series, 670, eds. Burdik C., Navratil O., Posta S., IOP Publishing Ltd, 2016, UNSP 012055  crossref  isi  scopus
    4. K. Zheltukhin, N. Zheltukhina, “Semi-Discrete Hyperbolic Equations Admitting Five Dimensional Characteristic X-Ring”, J. Nonlinear Math. Phys., 23:3 (2016), 351–367  crossref  mathscinet  isi  scopus
    5. Ismagil Habibullin, Mariya Poptsova, “Classification of a Subclass of Two-Dimensional Lattices via Characteristic Lie Rings”, SIGMA, 13 (2017), 073, 26 pp.  mathnet  crossref
    6. K. Zheltukhin, N. Zheltukhina, E. Bilen, “On a Class of Darboux-Integrable Semidiscrete Equations”, Adv. Differ. Equ., 2017, 182  crossref  mathscinet  isi  scopus
    7. M. N. Poptsova, I. T. Habibullin, “Algebraic properties of quasilinear two-dimensional lattices connected with integrability”, Ufa Math. J., 10:3 (2018), 86–105  mathnet  crossref  isi
    8. D. V. Millionshchikov, “Polynomial Lie algebras and growth of their finitely generated Lie subalgebras”, Proc. Steklov Inst. Math., 302 (2018), 298–314  mathnet  crossref  crossref  mathscinet  isi  elib
    9. Millionshchikov D., “Lie Algebras of Slow Growth and Klein-Gordon Pde”, Algebr. Represent. Theory, 21:5 (2018), 1037–1069  crossref  mathscinet  zmath  isi  scopus
    10. Kaptsov O.V., “Intermediate Systems and Higher-Order Differential Constraints”, J. Sib. Fed. Univ.-Math. Phys., 11:5 (2018), 550–560  mathnet  crossref  mathscinet  isi  scopus
    11. Zheltukhin K., Zheltukhina N., “On the Discretization of Laine Equations”, J. Nonlinear Math. Phys., 25:1 (2018), 166–177  crossref  mathscinet  isi
    12. Ufa Math. J., 11:3 (2019), 109–131  mathnet  crossref  isi
    13. 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  mathnet  crossref  crossref  isi  elib
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