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Hirota difference equation and a commutator identity on an associative algebra

Author: A. K. Pogrebkov
Original publication: Algebra i Analiz, tom 22 (2010), nomer 3.
Journal: St. Petersburg Math. J. 22 (2011), 473-483
MSC (2010): Primary 37K15
Published electronically: March 18, 2011
MathSciNet review: 2729946
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Abstract | References | Similar Articles | Additional Information

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.

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Additional Information

A. K. Pogrebkov
Affiliation: Steklov Mathematical Institute, Moscow 119991, Russia

Keywords: Hirota difference equation, commutator identity, extended operators, direct and inverse problems
Received by editor(s): February 19, 2010
Published electronically: March 18, 2011
Additional Notes: Supported in part by RFBR (grants 09-01-12150 and 09-01-93106), by Scientific Schools Program, and by the RAS program “Fundamental Problems of Nonlinear Dynamics”
Dedicated: To Ludwig Dmitrievich Faddeev on the occasion of his 75th birthday
Article copyright: © Copyright 2011 American Mathematical Society