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 SIGMA, 2012, Volume 8, 062, 33 pages (Mi sigma739)

Affine and finite Lie algebras and integrable Toda field equations on discrete space-time

Rustem Garifullina, Ismagil Habibullina, Marina Yangubaevab

a Ufa Institute of Mathematics, Russian Academy of Science, 112 Chernyshevskii Str., Ufa, 450077, Russia
b Faculty of Physics and Mathematics, Birsk State Social Pedagogical Academy, 10 Internationalnaya Str., Birsk, 452452, Russia

Abstract: Difference-difference systems are suggested corresponding to the Cartan matrices of any simple or affine Lie algebra. In the cases of the algebras $A_N$, $B_N$, $C_N$, $G_2$, $D_3$, $A_1^{(1)}$, $A_2^{(2)}$, $D^{(2)}_N$ these systems are proved to be integrable. For the systems corresponding to the algebras $A_2$, $A_1^{(1)}$, $A_2^{(2)}$ generalized symmetries are found. For the systems $A_2$, $B_2$, $C_2$, $G_2$, $D_3$ complete sets of independent integrals are found. The Lax representation for the difference-difference systems corresponding to $A_N$, $B_N$, $C_N$, $A^{(1)}_1$, $D^{(2)}_N$ are presented.

Keywords: affine Lie algebra; difference-difference systems; $S$-integrability; Darboux integrability; Toda field theory; integral; symmetry; Lax pair.

DOI: https://doi.org/10.3842/SIGMA.2012.062

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ArXiv: 1109.1689
MSC: 35Q53; 37K40
Received: April 24, 2012; in final form September 14, 2012; Published online September 18, 2012
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Citation: Rustem Garifullin, Ismagil Habibullin, Marina Yangubaeva, “Affine and finite Lie algebras and integrable Toda field equations on discrete space-time”, SIGMA, 8 (2012), 062, 33 pp.

Citation in format AMSBIB
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\by Rustem Garifullin, Ismagil Habibullin, Marina Yangubaeva
\paper Affine and finite Lie algebras and integrable Toda field equations on discrete space-time
\jour SIGMA
\yr 2012
\vol 8
\papernumber 062
\totalpages 33
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\crossref{https://doi.org/10.3842/SIGMA.2012.062}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2988032}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84867029853}

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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. I. T. Habibullin, M. V. Yangubaeva, “Formal diagonalization of a discrete Lax operator and conservation laws and symmetries of dynamical systems”, Theoret. and Math. Phys., 177:3 (2013), 1655–1679
2. M. V. Yangubaeva, “On structure of integrals for systems of discrete equations”, Ufa Math. J., 6:1 (2014), 111–116
3. S. V. Smirnov, “Darboux integrability of discrete two-dimensional Toda lattices”, Theoret. and Math. Phys., 182:2 (2015), 189–210
4. Habibullin I.T., Poptsova M.N., “Asymptotic Diagonalization of the Discrete Lax Pair Around Singularities and Conservation Laws For Dynamical Systems”, J. Phys. A-Math. Theor., 48:11 (2015), 115203
5. Demskoi D.K., Tran D.T., “Darboux integrability of determinant and equations for principal minors”, Nonlinearity, 29:7 (2016), 1973–1991
6. I. T. Habibullin, A. R. Khakimova, “Invariant manifolds and Lax pairs for integrable nonlinear chains”, Theoret. and Math. Phys., 191:3 (2017), 793–810
7. I. T. Habibullin, A. R. Khakimova, “On a method for constructing the Lax pairs for integrable models via a quadratic ansatz”, J. Phys. A-Math. Theor., 50:30 (2017), 305206
8. W. Fu, “Direct linearisation of the discrete-time two-dimensional Toda lattices”, J. Phys. A-Math. Theor., 51:33 (2018), 334001
9. I. T. Habibullin, A. R. Khakimova, “A direct algorithm for constructing recursion operators and Lax pairs for integrable models”, Theoret. and Math. Phys., 196:2 (2018), 1200–1216
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