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 TMF, 2000, Volume 122, Number 1, Pages 88–101 (Mi tmf557)

Integrable ordinary differential equations on free associative algebras

A. V. Mikhailovab, V. V. Sokolovc

a L. D. Landau Institute for Theoretical Physics, Russian Academy of Sciences
b University of Leeds
c Landau Institute for Theoretical Physics, Centre for Non-linear Studies

Abstract: We consider a classification problem for integrable nonlinear ordinary differential equations with an independent variable belonging to a free associative algebra $\mathcal M$. Every equation of this type admits an $m\times m$ matrix reduction for an arbitrary $m$. The existence of symmetries or first integrals belonging to $\mathcal M$ is used as an integrability criterion.

DOI: https://doi.org/10.4213/tmf557

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English version:
Theoretical and Mathematical Physics, 2000, 122:1, 72–83

Bibliographic databases:

Citation: A. V. Mikhailov, V. V. Sokolov, “Integrable ordinary differential equations on free associative algebras”, TMF, 122:1 (2000), 88–101; Theoret. and Math. Phys., 122:1 (2000), 72–83

Citation in format AMSBIB
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This publication is cited in the following articles:
1. I. Z. Golubchik, V. V. Sokolov, “One More Kind of the Classical Yang–Baxter Equation”, Funct. Anal. Appl., 34:4 (2000), 296–298
2. S. B. Leble, “Covariance of Lax Pairs and Integrability of the Compatibility Condition”, Theoret. and Math. Phys., 128:1 (2001), 890–905
3. Ustinov, NV, “Darboux integration of i (rho)over-dot = [H, f(rho)]”, Physics Letters A, 279:5–6 (2001), 333
4. Cieslinski, JL, “Darboux covariant equations of von Neumann type and their generalizations”, Journal of Mathematical Physics, 44:4 (2003), 1763
5. Ustinov NV, “The lattice equations of the Toda type with an interaction between a few neighbourhoods”, Journal of Physics A-Mathematical and General, 37:5 (2004), 1737–1746
6. S. B. Leble, “Necessary Covariance Conditions for a One-Field Lax Pair”, Theoret. and Math. Phys., 144:1 (2005), 985–994
7. Leble S., “Covariant forms of Lax one-field operators: From abelian to noncommutative”, Bilinear Integrable Systems: From Classical to Quatum, Continuous to Discrete, Nato Science Series, Series II: Mathematics, Physics and Chemistry, 201, 2006, 161–173
8. Calogero F., “An integrable many-body problem”, J Math Phys, 52:10 (2011), 102702
9. Calogero F., “Two Quite Similar Matrix ODEs and the Many-Body Problems Related to Them”, Int. J. Geom. Methods Mod. Phys., 9:2, SI (2012), 1260002
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