Communications in Mathematical Physics

, Volume 211, Issue 1, pp 231–251 | Cite as

Integrable ODEs on Associative Algebras

  • A. V. Mikhailov
  • V. V. Sokolov


In this paper we give definitions of basic concepts such as symmetries, first integrals, Hamiltonian and recursion operators suitable for ordinary differential equations on associative algebras, and in particular for matrix differential equations. We choose existence of hierarchies of first integrals and/or symmetries as a criterion for integrability and justify it by examples. Using our componentless approach we have solved a number of classification problems for integrable equations on free associative algebras. Also, in the simplest case, we have listed all possible Hamiltonian operators of low order.


Differential Equation Integrable Equation Ordinary Differential Equation Basic Concept Classification Problem 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • A. V. Mikhailov
    • 1
  • V. V. Sokolov
    • 2
  1. 1.Applied Math. Department, University of Leeds, Leeds, LS2 9JT, UKUK
  2. 2.Centre for Nonlinear Studies, at Landau Institute for Theoretical Physics, Russian Academy of Sciences,¶2 Kosygina st., Moscow, 117940, RussiaRU

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