Communications in Mathematical Physics

, Volume 193, Issue 2, pp 245–268 | Cite as

Integrable Evolution Equations on Associative Algebras

  • Peter J. Olver
  • Vladimir V. Sokolov


This paper surveys the classification of integrable evolution equations whose field variables take values in an associative algebra, which includes matrix, Clifford, and group algebra valued systems. A variety of new examples of integrable systems possessing higher order symmetries are presented. Symmetry reductions lead to an associative algebra-valued version of the Painlevé transcendent equations. The basic theory of Hamiltonian structures for associative algebra-valued systems is developed and the biHamiltonian structures for several examples are found.


Evolution Equation Integrable System Basic Theory Field Variable Associative Algebra 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Peter J. Olver
    • 1
  • Vladimir V. Sokolov
    • 2
  1. 1.School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA.¶E-mail:,∼olverUS
  2. 2.Ufa Mathematical Institute, Russian Academy of Sciences, Chernyshevski str. 112, 450000, Ufa, Russia. E-mail: sokolov@imat.rb.ruRU

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