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

DOI:
https://doi.org/10.1090/S1061-0022-2011-01153-7

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

Email:
pogreb@mi.ras.ru

DOI:
https://doi.org/10.1090/S1061-0022-2011-01153-7

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