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TMF, 2008, Volume 154, Number 3, Pages 477–491 (Mi tmf6182)  

This article is cited in 10 scientific papers (total in 10 papers)

Commutator identities on associative algebras and the integrability of nonlinear evolution equations

A. K. Pogrebkov

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: We show that commutator identities on associative algebras generate solutions of the linearized versions of integrable equations. In addition, we introduce a special dressing procedure in a class of integral operators that allows deriving both the nonlinear integrable equation itself and its Lax pair from such a commutator identity. The problem of constructing new integrable nonlinear evolution equations thus reduces to the problem of constructing commutator identities on associative algebras.

Keywords: nonlinear evolution equation, Lax pair

DOI: https://doi.org/10.4213/tmf6182

Full text: PDF file (482 kB)
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English version:
Theoretical and Mathematical Physics, 2008, 154:3, 405–417

Bibliographic databases:


Citation: A. K. Pogrebkov, “Commutator identities on associative algebras and the integrability of nonlinear evolution equations”, TMF, 154:3 (2008), 477–491; Theoret. and Math. Phys., 154:3 (2008), 405–417

Citation in format AMSBIB
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  • http://mi.mathnet.ru/eng/tmf/v154/i3/p477

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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. St. Petersburg Math. J., 22:3 (2011), 473–483  mathnet  crossref  mathscinet  zmath  isi
    2. A. K. Pogrebkov, “Hirota difference equation: Inverse scattering transform, Darboux transformation, and solitons”, Theoret. and Math. Phys., 181:3 (2014), 1585–1598  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    3. A. K. Pogrebkov, “Commutator identities on associative algebras, the non-Abelian Hirota difference equation and its reductions”, Theoret. and Math. Phys., 187:3 (2016), 823–834  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
    4. Ismagil Habibullin, Mariya Poptsova, “Classification of a Subclass of Two-Dimensional Lattices via Characteristic Lie Rings”, SIGMA, 13 (2017), 073, 26 pp.  mathnet  crossref
    5. 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
    6. A. K. Pogrebkov, “Higher Hirota difference equations and their reductions”, Theoret. and Math. Phys., 197:3 (2018), 1779–1796  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    7. Ufa Math. J., 11:3 (2019), 109–131  mathnet  crossref  isi
    8. 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  mathscinet  isi  elib
    9. A. K. Pogrebkov, “Commutator identities and integrable hierarchies”, Theoret. and Math. Phys., 205:3 (2020), 1585–1592  mathnet  crossref  crossref  mathscinet  isi  elib
    10. Habibullin I.T. Kuznetsova M.N. Sakieva A.U., “Integrability Conditions For Two-Dimensional Toda-Like Equations”, J. Phys. A-Math. Theor., 53:39 (2020), 395203  crossref  mathscinet  isi
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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