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 Algebra i Analiz, 2010, Volume 22, Issue 3, Pages 191–205 (Mi aa1192)

Research Papers

Hirota difference equation and a commutator identity on an associative algebra

A. K. Pogrebkov

Steklov Mathematical Institute, Moscow, Russia

Abstract: In earlier papers of the author it was shown that some simple commutator identities on an associative algebra generate integrable nonlinear equations. Here, this observation is generalized to the case of difference nonlinear equations. The identity under study leads, under a special realization of the elements of the associative algebra, to the famous Hirota difference equation. Direct and inverse problems are considered for this equation, as well as some properties of its solutions. Finally, some other commutator identities are discussed and their relationship with integrable nonlinear equations, both differential and difference, is demonstrated.

Keywords: Hirota difference equation, commutator identity, extended operators, direct and inverse problems.

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English version:
St. Petersburg Mathematical Journal, 2011, 22:3, 473–483

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Citation: A. K. Pogrebkov, “Hirota difference equation and a commutator identity on an associative algebra”, Algebra i Analiz, 22:3 (2010), 191–205; St. Petersburg Math. J., 22:3 (2011), 473–483

Citation in format AMSBIB
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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. A. K. Pogrebkov, “Hirota difference equation: Inverse scattering transform, Darboux transformation, and solitons”, Theoret. and Math. Phys., 181:3 (2014), 1585–1598
2. 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
3. Andrei K. Pogrebkov, “Symmetries of the Hirota Difference Equation”, SIGMA, 13 (2017), 053, 14 pp.
4. A. K. Pogrebkov, “Higher Hirota difference equations and their reductions”, Theoret. and Math. Phys., 197:3 (2018), 1779–1796
5. Pogrebkov A., “Hirota Difference Equation and Darboux System: Mutual Symmetry”, Symmetry-Basel, 11:3 (2019), 436
6. A. K. Pogrebkov, “Commutator identities and integrable hierarchies”, Theoret. and Math. Phys., 205:3 (2020), 1585–1592
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