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

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Teoreticheskaya i Matematicheskaya Fizika, 2008, Volume 154, Number 3, Pages 477–491
DOI: https://doi.org/10.4213/tmf6182 (Mi tmf6182)

This article is cited in 16 scientific papers (total in 16 papers)

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

A. K. Pogrebkov

Steklov Mathematical Institute, Russian Academy of Sciences

Full-text PDF (482 kB) Citations (16)

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

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.

English version:
Theoretical and Mathematical Physics, 2008, Volume 154, Issue 3, Pages 405–417
DOI: https://doi.org/10.1007/s11232-008-0035-6

Bibliographic databases:

Document Type: Article

Language: Russian

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

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Linking options:

https://www.mathnet.ru/eng/tmf6182

https://doi.org/10.4213/tmf6182

https://www.mathnet.ru/eng/tmf/v154/i3/p477

This publication is cited in the following 16 articles:

Ge Yi, Rong Hu, Kelei Tian, Ying Xu, “On the dispersionless Davey-Stewartson hierarchy: the tau function, the Riemann-Hilbert problem and the Hamilton-Jacobi theory”, Phys. Scr., 99:5 (2024), 055243
A. K. Pogrebkov, “Equation with a lower negative time number in the Davey–Stewartson hierarchy”, Theoret. and Math. Phys., 221:3 (2024), 2022–2030
A. K. Pogrebkov, “Nonexplicit versions of integrable equations”, Theoret. and Math. Phys., 217:3 (2023), 1907–1913
Xue-Jiao He, Xing Lü, “M-lump solution, soliton solution and rational solution to a (3+1)-dimensional nonlinear model”, Mathematics and Computers in Simulation, 197 (2022), 327
Andrei K. Pogrebkov, “Negative Times of the Davey–Stewartson Integrable Hierarchy”, SIGMA, 17 (2021), 091, 12 pp.
Pogrebkov A.K., “Kadomtsev-Petviashvili Hierarchy: Negative Times”, Mathematics, 9:16 (2021), 1988
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
A. K. Pogrebkov, “Commutator identities and integrable hierarchies”, Theoret. and Math. Phys., 205:3 (2020), 1585–1592
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
Ufa Math. J., 11:3 (2019), 109–131
M. N. Poptsova, I. T. Habibullin, “Algebraic properties of quasilinear two-dimensional lattices connected with integrability”, Ufa Math. J., 10:3 (2018), 86–105
A. K. Pogrebkov, “Higher Hirota difference equations and their reductions”, Theoret. and Math. Phys., 197:3 (2018), 1779–1796
Ismagil Habibullin, Mariya Poptsova, “Classification of a Subclass of Two-Dimensional Lattices via Characteristic Lie Rings”, SIGMA, 13 (2017), 073, 26 pp.
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
A. K. Pogrebkov, “Hirota difference equation: Inverse scattering transform, Darboux transformation, and solitons”, Theoret. and Math. Phys., 181:3 (2014), 1585–1598
St. Petersburg Math. J., 22:3 (2011), 473–483

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

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