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 Izv. Akad. Nauk SSSR Ser. Mat., 1990, Volume 54, Issue 2, Pages 258–274 (Mi izv1093)

A theorem on two commuting automorphisms, and integrable differential equations

O. I. Bogoyavlenskii

Abstract: Constructions are found for differential equations in an arbitrary continuous associative algebra $\mathfrak A$ that admit an equivalent Lax representation (with spectral parameter) in the space of linear operators acting on $\mathfrak A$. The constructions use commuting automorphisms of $\mathfrak A$. Applications of the main construction are indicated for the construction of integrable Euler equations in the direct sum of the Lie algebras $\operatorname{gl}(n,R)$ and $\operatorname{so}(n,R)$. Constructions are presented for matrix differential equations admitting a Lax representation with several spectral parameters.

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English version:
Mathematics of the USSR-Izvestiya, 1991, 36:2, 263–279

Bibliographic databases:

UDC: 539.2
MSC: Primary 58F07; Secondary 35Q20, 70H05

Citation: O. I. Bogoyavlenskii, “A theorem on two commuting automorphisms, and integrable differential equations”, Izv. Akad. Nauk SSSR Ser. Mat., 54:2 (1990), 258–274; Math. USSR-Izv., 36:2 (1991), 263–279

Citation in format AMSBIB
\Bibitem{Bog90} \by O.~I.~Bogoyavlenskii \paper A~theorem on two commuting automorphisms, and integrable differential equations \jour Izv. Akad. Nauk SSSR Ser. Mat. \yr 1990 \vol 54 \issue 2 \pages 258--274 \mathnet{http://mi.mathnet.ru/izv1093} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1062513} \zmath{https://zbmath.org/?q=an:0716.58018|0699.58041} \adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?1991IzMat..36..263B} \transl \jour Math. USSR-Izv. \yr 1991 \vol 36 \issue 2 \pages 263--279 \crossref{https://doi.org/10.1070/IM1991v036n02ABEH002021} 

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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. O. I. Bogoyavlenskii, “Breaking solitons in $2+1$-dimensional integrable equations”, Russian Math. Surveys, 45:4 (1990), 1–89
2. O. I. Bogoyavlenskii, “Algebraic constructions of integrable dynamical systems-extensions of the Volterra system”, Russian Math. Surveys, 46:3 (1991), 1–64
3. O. I. Bogoyavlenskii, “Euler equations on finite-dimensional Lie coalgebras, arising in problems of mathematical physics”, Russian Math. Surveys, 47:1 (1992), 117–189
4. Yoshiaki Itoh, “A combinatorial method for the vanishing of the Poisson brackets of an integrable Lotka–Volterra system”, J. Phys. A: Math. Theor, 42:2 (2009), 025201
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