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


Symmetries of \(\mathbb {E}\subset J^\infty (n,m)\) are vector fields that preserve solutions of \(\mathbb {E}\). Effectively, this means that they preserve the Cartan distribution on the equation at hand. We discuss symmetries and related notions in this chapter and describe solutions to Problems  1.5,  1.13, and  1.14.


  1. 4.
    Andrews, G.E., Askey, R., Roy, R.: Special Functions. Encyclopedia of Mathematics and its Applications, vol. 71. Cambridge University Press, Cambridge (1999). ISBN:978-0-521-62321-6Google Scholar
  2. 9.
    Baran, H., Krasilshchik, I.S., Morozov, O.I., Vojčák, P.: Higher symmetries of cotangent coverings for Lax-integrable multi-dimensional partial differential equations and lagrangian deformations. In: Konopelchenko, B.G., et al. (eds.) Physics and Mathematics of Nonlinear Phenomena 2013. Journal of Physics: Conference Series, vol. 482, p. 012002 (2014). arXiv:1309.7435Google Scholar
  3. 10.
    Baran, H., Krasilshchik, I.S., Morozov, O.I., Vojčák, P.: Symmetry reductions and exact solutions of Lax integrable 3-dimensional systems. J. Nonlinear Math. Phys. 21(4), 643–671 (2014). arXiv:1407.0246Google Scholar
  4. 13.
    Baran, H., Krasilshchik, I.S., Morozov, O.I., Vojčák, P.: Coverings over Lax integrable equations and their nonlocal symmetries. Theor. Math. Phys. 188(3), 1273–1295 (2016). arXiv:1507.00897Google Scholar
  5. 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
  6. 18.
    Bocharov, A.V., Chetverikov, V.N., Duzhin, S.V., Khorkova, N.G., Krasilshchik, I.S., Samokhin, A.V., Torkhov, Y.N., Verbovetsky, A.M., Vinogradov, A.M.: In: Krasilshchik, I.S., Vinogradov, A.M. (eds.) Symmetries and Conservation Laws for Differential Equations of Mathematical Physics. Monograph. American Mathematical Society, Providence (1999)Google Scholar
  7. 29.
    Dunajski, M., Kryński, W.: Einstein-Weyl geometry, dispersionless Hirota equation and Veronese webs. Math. Proc. Camb. Philos. Soc. 157(1), 139–150 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 37.
    Fuchssteiner, B.: Mastersymmetries, higher order time-dependent symmetries and conserved densities of nonlinear evolution equations. Prog. Theor. Phys. 70(6), 1508–1522 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 38.
    Fuchssteiner, B., Fokas, A.S.: Symplectic structures, their Bäcklund transformations and hereditary symmetries. Phys. D 4(1), 47–66 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 48.
    Ibragimov, N.: Transformation Groups Applied to Mathematical Physics. Reidel, Dordrecht (1985)CrossRefzbMATHGoogle Scholar
  11. 66.
    Khorkova, N.G.: Conservation laws and nonlocal symmetries. Math. Notes 44, 562–568 (1989)Google Scholar
  12. 75.
    Krasilshchik, I.S., Lychagin, V.V., Vinogradov, A.M.: Geometry of Jet Spaces and Nonlinear Partial Differential Equations. Gordon and Breach, New York (1986)Google Scholar
  13. 112.
    Pavlov, M.V.: Integrable hydrodynamic chains. J. Math. Phys. 44, 4134–4156 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 140.
    Vinogradov, A.M., Krasilshchik, I.S.: A method of computing higher symmetries of nonlinear evolution equations and nonlocal symmetries. Sov. Math. Dokl. 22, 235–239 (1980)Google 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