Recursion Operators for Symmetries

  • Joseph Krasil’shchik
  • Alexander Verbovetsky
  • Raffaele Vitolo
Part of the Texts & Monographs in Symbolic Computation book series (TEXTSMONOGR)


A recursion operator for symmetries of an equation \(\mathcal {E}\) is a \(\mathcal {C}\)-differential operator \(\mathcal {R}\colon \varkappa =\mathcal {F}(\mathcal {E};m) \to \varkappa \) that takes symmetries of \(\mathcal {E}\) to themselves. We expose below a computational theory of such operators based on the tangent covering techniques. The simplest version of this theory relates to local operators, but in reality all recursion operators, except for the case of linear equations with constant coefficients, are nonlocal. Such operators, in general, act on shadows of symmetries only. Unfortunately, to the best of our knowledge, a self-contained theory for these operators (as well as for nonlocal operators of other types that are considered below) does not exist at the moment, but some reasonable ideas can be applied to particular classes of examples nevertheless. In this chapter, we give the solution to Problems  1.20 and  1.28.


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© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  • Joseph Krasil’shchik
    • 1
  • Alexander Verbovetsky
    • 2
  • Raffaele Vitolo
    • 3
  1. 1.V.A. Trapeznikov Institute of Control Sciences RASIndependent University of MoscowMoscowRussia
  2. 2.Independent University of MoscowMoscowRussia
  3. 3.Department of Mathematics and Physics ‘E. De Giorgi’University of SalentoLecceItaly

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