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

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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UDC: 517.957

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
\Bibitem{ZhiMurHab12} \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 \mathnet{http://mi.mathnet.ru/ufa156} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3429920} 

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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. Habibullin I., “Characteristic Lie Rings, Finitely-Generated Modules and Integrability Conditions for (2+1)-Dimensional Lattices”, Phys. Scr., 87:6 (2013), 065005
2. V. M. Zhuravlev, “Matrichnye funktsionalnye podstanovki dlya integriruemykh dinamicheskikh sistem i uravneniya Landau–Lifshitsa”, Nelineinaya dinam., 10:1 (2014), 35–48
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
4. K. Zheltukhin, N. Zheltukhina, “Semi-Discrete Hyperbolic Equations Admitting Five Dimensional Characteristic X-Ring”, J. Nonlinear Math. Phys., 23:3 (2016), 351–367
5. Ismagil Habibullin, Mariya Poptsova, “Classification of a Subclass of Two-Dimensional Lattices via Characteristic Lie Rings”, SIGMA, 13 (2017), 073, 26 pp.
6. K. Zheltukhin, N. Zheltukhina, E. Bilen, “On a Class of Darboux-Integrable Semidiscrete Equations”, Adv. Differ. Equ., 2017, 182
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
8. D. V. Millionshchikov, “Polynomial Lie algebras and growth of their finitely generated Lie subalgebras”, Proc. Steklov Inst. Math., 302 (2018), 298–314
9. Millionshchikov D., “Lie Algebras of Slow Growth and Klein-Gordon Pde”, Algebr. Represent. Theory, 21:5 (2018), 1037–1069
10. Kaptsov O.V., “Intermediate Systems and Higher-Order Differential Constraints”, J. Sib. Fed. Univ.-Math. Phys., 11:5 (2018), 550–560
11. Zheltukhin K., Zheltukhina N., “On the Discretization of Laine Equations”, J. Nonlinear Math. Phys., 25:1 (2018), 166–177
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