On integrability of the Kontsevich non-abelian ODE system

Please always quote using this URN: urn:nbn:de:0297-zib-13113
  • We consider systems of ODEs with the right hand side being Laurent polynomials in several non-commutative unknowns. In particular, these unknowns could be matrices of arbitrary size. An important example of such a system was proposed by M. Kontsevich. We prove the integrability of the Kontsevich system by finding a Lax pair, corresponding first integrals and commuting flows. We also provide a pre-Hamiltonian operator which maps gradients of integrals for the Kontsevich system to symmetries.

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Author:Thomas Wolf, Olga Efimovskaya
Document Type:ZIB-Report
Tag:integrability, Lax pairs, noncommutative ODE, Laurent ODE
MSC-Classification:34-XX ORDINARY DIFFERENTIAL EQUATIONS
37-XX DYNAMICAL SYSTEMS AND ERGODIC THEORY [See also 26A18, 28Dxx, 34Cxx, 34Dxx, 35Bxx, 46Lxx, 58Jxx, 70-XX]
Date of first Publication:2011/06/20
Series (Serial Number):ZIB-Report (11-24)
ArXiv Id:http://arxiv.org/abs/1108.4208v1
Published in:Appeared in: Lett. in Math. Phys., vol 100, no 2 (2012), p 161-170
DOI:https://doi.org/10.1007/s11005-011-0527-4