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

Full-text PDF Free Access

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.

**1.**A. K. Pogrebkov,*Commutator identities on associative algebras and the integrability of nonlinear evolution equations*, Teoret. Mat. Fiz.**154**(2008), no. 3, 477–491 (Russian, with Russian summary); English transl., Theoret. and Math. Phys.**154**(2008), no. 3, 405–417. MR**2431558**, https://doi.org/10.1007/s11232-008-0035-6**2.**A. K. Pogrebkov,*2D Toda chain and associated commutator identity*, Geometry, topology, and mathematical physics, Amer. Math. Soc. Transl. Ser. 2, vol. 224, Amer. Math. Soc., Providence, RI, 2008, pp. 261–269. MR**2462365**, https://doi.org/10.1090/trans2/224/13**3.**M. Boiti, F. Pempinelli, A. K. Pogrebkov, and M. C. Polivanov,*Resolvent approach for two-dimensional scattering problems. Application to the nonstationary Schrödinger problem and KPI equation*, Teoret. Mat. Fiz.**93**(1992), no. 2, 181–210 (English, with English and Russian summaries); English transl., Theoret. and Math. Phys.**93**(1992), no. 2, 1200–1224 (1993). MR**1233541**, https://doi.org/10.1007/BF01083519**4.**M. Boiti, F. Pempinelli, A. K. Pogrebkov, and M. C. Polivanov,*Resolvent approach for the nonstationary Schrödinger equation*, Inverse Problems**8**(1992), no. 3, 331–364. MR**1166486****5.**M. Boiti, F. Pempinelli, and A. Pogrebkov,*Properties of solutions of the Kadomtsev-Petviashvili I equation*, J. Math. Phys.**35**(1994), no. 9, 4683–4718. MR**1290895**, https://doi.org/10.1063/1.530808**6.**M. Boiti, F. Pempinelli, A. K. Pogrebkov, and B. Prinari,*Extended resolvent and inverse scattering with an application to KPI*, J. Math. Phys.**44**(2003), no. 8, 3309–3340. Integrability, topological solitons and beyond. MR**2006753**, https://doi.org/10.1063/1.1587874**7.**Etsurō Date, Masaki Kashiwara, Michio Jimbo, and Tetsuji Miwa,*Transformation groups for soliton equations*, Nonlinear integrable systems—classical theory and quantum theory (Kyoto, 1981) World Sci. Publishing, Singapore, 1983, pp. 39–119. MR**725700****8.**A. Yu. Orlov and E. I. Schulman,*Additional symmetries for integrable equations and conformal algebra representation*, Lett. Math. Phys.**12**(1986), no. 3, 171–179. MR**865754**, https://doi.org/10.1007/BF00416506**9.**A. Yu. Orlov,*Vertex operator, \overline∂-problem, symmetries, variational identities and Hamiltonian formalism for 2+1 integrable systems*, Plasma theory and nonlinear and turbulent processes in physics, Vol. 1, 2 (Kiev, 1987) World Sci. Publishing, Singapore, 1988, pp. 116–134. MR**957158****10.**V. E. Zaharov and A. B. Šabat,*A plan for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I*, Funkcional. Anal. i Priložen.**8**(1974), no. 3, 43–53 (Russian). MR**0481668****11.**A. K. Pogrebkov,*On time evolutions associated with the nonstationary Schrödinger equation*, L. D. Faddeev’s Seminar on Mathematical Physics, Amer. Math. Soc. Transl. Ser. 2, vol. 201, Amer. Math. Soc., Providence, RI, 2000, pp. 239–255. MR**1772293**, https://doi.org/10.1090/trans2/201/13**12.**Ryogo Hirota,*Nonlinear partial difference equations. I. A difference analogue of the Korteweg-de Vries equation*, J. Phys. Soc. Japan**43**(1977), no. 4, 1424–1433. MR**0460934**, https://doi.org/10.1143/JPSJ.43.1424

Ryogo Hirota,*Nonlinear partial difference equations. II. Discrete-time Toda equation*, J. Phys. Soc. Japan**43**(1977), no. 6, 2074–2078. MR**0460935**, https://doi.org/10.1143/JPSJ.43.2074

Ryogo Hirota,*Nonlinear partial difference equations. III. Discrete sine-Gordon equation*, J. Phys. Soc. Japan**43**(1977), no. 6, 2079–2086. MR**0460936**, https://doi.org/10.1143/JPSJ.43.2079**13.**Ryogo Hirota,*Discrete analogue of a generalized Toda equation*, J. Phys. Soc. Japan**50**(1981), no. 11, 3785–3791. MR**638804**, https://doi.org/10.1143/JPSJ.50.3785**14.**A. V. Zabrodin,*Hirota difference equations*, Teoret. Mat. Fiz.**113**(1997), no. 2, 179–230 (Russian, with English and Russian summaries); English transl., Theoret. and Math. Phys.**113**(1997), no. 2, 1347–1392 (1998). MR**1608967**, https://doi.org/10.1007/BF02634165**15.**I. Krichever, P. Wiegmann, and A. Zabrodin,*Elliptic solutions to difference non-linear equations and related many-body problems*, Comm. Math. Phys.**193**(1998), no. 2, 373–396. MR**1618143**, https://doi.org/10.1007/s002200050333**16.**L. V. Bogdanov and B. G. Konopelchenko,*Generalized hierarchy: Möbius symmetry, symmetry constraints and Calogero-Moser system*, solv-int/9912005 (1999).**17.**A. Zabrodin,*Bäcklund transformations and Hirota equation and supersymmetric Bethe ansatz*,`arXiv:0705.4006v1`(2007).**18.**S. V. Manakov,*The inverse scattering transform for the time-dependent Schrödinger equation and Kadomtsev-Petviashvili equation*, Physica D**3**(1981), 420-427.

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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