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TMF, 2013, Volume 177, Number 3, Pages 441–467 (Mi tmf8581)  

This article is cited in 7 scientific papers (total in 7 papers)

Formal diagonalization of a discrete Lax operator and conservation laws and symmetries of dynamical systems

I. T. Habibullin, M. V. Yangubaeva

Institute of Mathematics with Computing Center, Ufa Science Center, RAS, Ufa, Russia

Abstract: We consider the problem of constructing a formal asymptotic expansion in the spectral parameter for an eigenfunction of a discrete linear operator. We propose a method for constructing an expansion that allows obtaining conservation laws of discrete dynamical systems associated with a given linear operator. As illustrative examples, we consider known nonlinear models such as the discrete potential Korteweg–de Vries equation, the discrete version of the derivative nonlinear Schrödinger equation, the Veselov–Shabat dressing chain, and others. We describe the infinite set of conservation laws for the discrete Toda chain corresponding to the Lie algebra $A_1^{(1)}$. We find new examples of integrable systems of equations on a square lattice.

Keywords: Lax pair, asymptotic expansion, conservation law, symmetry, equations on a quad graph, discrete nonlinear Schrödinger equation, dressing method

DOI: https://doi.org/10.4213/tmf8581

Full text: PDF file (552 kB)
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English version:
Theoretical and Mathematical Physics, 2013, 177:3, 1655–1679

Bibliographic databases:

Received: 24.07.2013
Revised: 16.08.2013

Citation: I. T. Habibullin, M. V. Yangubaeva, “Formal diagonalization of a discrete Lax operator and conservation laws and symmetries of dynamical systems”, TMF, 177:3 (2013), 441–467; Theoret. and Math. Phys., 177:3 (2013), 1655–1679

Citation in format AMSBIB
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  • http://mi.mathnet.ru/eng/tmf/v177/i3/p441

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

    This publication is cited in the following articles:
    1. R. N. Garifullin, A. V. Mikhailov, R. I. Yamilov, “Discrete equation on a square lattice with a nonstandard structure of generalized symmetries”, Theoret. and Math. Phys., 180:1 (2014), 765–780  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    2. I. T. Habibullin, M. N. Poptsova, “Asymptotic diagonalization of the discrete Lax pair around singularities and conservation laws for dynamical systems”, J. Phys. A-Math. Theor., 48:11 (2015), 115203  crossref  mathscinet  zmath  adsnasa  isi  scopus
    3. A. V. Mikhailov, “Formal diagonalisation of Lax–Darboux schemes”, Model. i analiz inform. sistem, 22:6 (2015), 795–817  mathnet  crossref  mathscinet  elib
    4. I. T. Habibullin, A. R. Khakimova, M. N. Poptsova, “On a method for constructing the Lax pairs for nonlinear integrable equations”, J. Phys. A-Math. Theor., 49:3 (2016), 035202  crossref  mathscinet  zmath  adsnasa  isi  scopus
    5. M. N. Poptsova, I. T. Habibullin, “Symmetries and conservation laws for a two-component discrete potentiated Korteweg–de Vries equation”, Ufa Math. J., 8:3 (2016), 109–121  mathnet  crossref  mathscinet  isi  elib
    6. S. Lou, Y. Shi, D.-J. Zhang, “Spectrum transformation and conservation laws of lattice potential KdV equation”, Front. Math. China, 12:2 (2017), 403–416  crossref  mathscinet  zmath  isi  scopus
    7. Ufa Math. J., 9:3 (2017), 158–164  mathnet  crossref  mathscinet  isi  elib  elib
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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