Dark equations and their light integrability

Denis Blackmore, Anatolij K. Prykarpatski

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11 Scopus citations


A relatively new approach to analyzing integrability, based upon differential-algebraic and symplectic techniques, is applied to some "dark equations "of the type introduced by Boris Kupershmidt. These dark equations have unusual properties and are not particularly well-understood. In particular, dark three-component polynomial Burgers type systems are studied in detail. Their matrix Lax representations are constructed, and the related symmetry recursion operators and infinite hierarchies of integrable nonlinear dynamical systems along with their Lax representations are derived. New linear Lax spectral problems for dark integrable countable hierarchies of dynamical systems are proposed and some special cases are considered as a means of indicating that the approach presented is applicable to a far wider class of dark equations than analyzed here.

Original languageEnglish (US)
Pages (from-to)407-428
Number of pages22
JournalJournal of Nonlinear Mathematical Physics
Issue number3
StatePublished - Jul 3 2014

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics


  • Burgers type system
  • Lax integrability
  • asymptotic analysis
  • commuting infinite hierarchies of dynamical systems
  • conserved quantities
  • differential-algebraic approach
  • symmetry recursion operator


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