Recursion Operators for Symmetries

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

Abstract

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.

References

  1. 14.
    Baran, H., Krasilshchik, I.S., Morozov, O.I., Vojčák, P.: Nonlocal symmetries of Lax integrable equations: a comparative study. Submitted to Theor. Math. Phys. (2016). arXiv:1611.04938Google Scholar
  2. 24.
    Dorfman, I.Y.: Dirac Structures and Integrability of Nonlinear Evolution Equations. Wiley, Chichester/New York (1993)MATHGoogle Scholar
  3. 61.
    Kersten, P., Krasilshchik, I., Verbovetsky, A.: A geometric study of the dispersionless Boussinesq type equation. Acta Appl. Math. 90, 143–178 (2006)Google Scholar
  4. 70.
    Krasilshchik, I.S.: Algebras with flat connections and symmetries of differential equations. In: Komrakov, B.P., Krasilshchik, I.S., Litvinov, G.L., Sossinsky, A.B. (eds.) Lie Groups and Lie Algebras: Their Representations, Generalizations and Applications, pp. 407–424. Kluwer, Dordrecht/Boston (1998)Google Scholar
  5. 79.
    Krasilshchik, J., Verbovetsky, A., Vitolo, R.: A unified approach to computation of integrable structures. Acta Appl. Math. 120(1), 199–218 (2012)Google Scholar
  6. 89.
    Maltsev, A.Y., Novikov, S.P.: On the local systems Hamiltonian in the weakly non-local Poisson brackets. Phys. D 156(1–2), 53–80 (2001)MathSciNetCrossRefMATHGoogle Scholar
  7. 90.
    Malykh, A.A., Nutku, Y., Sheftel, M.B.: Partner symmetries of the complex Monge–Ampère equation yield hyper-Kähler metrics without continuous symmetries. J. Phys. A 36(39), 10023 (2003). arXiv:math-ph/0305037Google Scholar
  8. 96.
    Marvan, M.: Another look on recursion operators. In: Differential Geometry and Applications. Proceedings of the Conference, Brno, 1995, pp. 393–402. Masaryk University, Brno (1996)Google Scholar
  9. 99.
    Marvan, M., Sergyeyev, A.: Recursion operators for dispersionless integrable systems in any dimension. Inverse Prob. 28, 025011 (2012). arXiv:nlin/1107.0784Google Scholar
  10. 102.
    Morozov, O.I.: Recursion operators and nonlocal symmetries for integrable rmdKP and rdDym equations (2012). arXiv:nlin/1202.2308Google Scholar
  11. 103.
    Morozov, O.I.: A recursion operator for the Universal Hierarchy equation via Cartan’s method of equivalence. Cent. Eur. J. Math. 12(2), 271–283 (2014)MathSciNetMATHGoogle Scholar
  12. 105.
    Neyzi, F., Nutku, Y., Sheftel, M.B.: Multi-hamiltonian structure of Plebanski’s second heavenly equation. J. Phys. A 38, 8473 (2005)MathSciNetCrossRefMATHGoogle Scholar
  13. 129.
    Sergyeyev, A.: Locality of symmetries generated by nonhereditary, inhomogeneous, and time-dependent recursion operators: a new application for formal symmetries. Acta Appl. Math. 83, 95–109 (2004)MathSciNetCrossRefMATHGoogle Scholar
  14. 130.
    Sergyeyev, A.: Why nonlocal recursion operators produce local symmetries: new results and applications. J. Phys. A 38(15), 3397–3407 (2005)MathSciNetCrossRefMATHGoogle Scholar
  15. 132.
    Sergyeyev, A.: Recursion operators for multidimensional integrable PDEs. (2017, Submitted). arXiv:1710.05907Google Scholar
  16. 133.
    Sergyeyev, A.: A simple construction of recursion operators for multidimensional dispersionless integrable systems. J. Math. Anal. Appl. 454, 468–480 (2017). arXiv:1501.01955Google Scholar
  17. 135.
    Sheftel, M.B., Malykh, A.A.: On classification of second-order PDEs possessing partner symmetries. J. Phys. A 42(39), 395202 (2009). arXiv:0904.2909Google Scholar
  18. 145.
    Wang, J.P.: Symmetries and Conservation Laws of Evolution Equations. PhD thesis, Vrije Universiteit/Thomas Stieltjes Institute, Amsterdam (1998)Google Scholar
  19. 146.
    Wang, J.P.: A list of 1 + 1 dimensional integrable equations and their properties. J. Nonlinear Math. Phys. 9, 213–233 (2002)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© 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

Personalised recommendations