Conservation Laws and Nonlocal Variables

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


We discuss here the notion of conservation laws and briefly the theory of Abelian coverings over infinitely prolonged equations. Computation of conservation laws is also closely related to that of cosymmetries , and we shall continue this discussion in Chap. 4 below. In this chapter we give the solution to Problems  1.7,  1.13, and  1.15 posed in Chap.  1.


  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. 30.
    Ferapontov, E.V.: Surfaces in 3-space possessing nontrivial deformations which preserve the shape operator. In: Integrable Systems in Differential Geometry, Tokyo, July 2000, Contemporary Mathematics, vol. 508, pp. 145–159. AMS, Providence (2002). arXiv:math/0107122Google Scholar
  3. 41.
    Gibbons, J., Tsarev, S.P.: Reductions of the Benney equations. Phys. Lett. A 211, 19–24 (1996)Google Scholar
  4. 42.
    Göktas, Ü., Hereman, W.: Symbolic computation of conserved densities for systems of nonlinear evolution equations. J. Symb. Comput. 24(5), 591–621 (1997)Google Scholar
  5. 43.
    Golovko, V.A., Kersten, P.H.M., Krasilshchik, I.S., Verbovetsky, A.M.: On integrability of the Camassa-Holm equation and its invariants. Acta Appl. Math. 101, 59–83 (2008)Google Scholar
  6. 46.
    Hereman, W., Adams, P.J., Eklund, H.L., Hickman, M.S., Herbst, B.M.: Direct methods and symbolic software for conservation laws of nonlinear equations. In: Yan, Z. (ed.) Advances in Nonlinear Waves and Symbolic Computation, chap. 2, pp. 19–79. Nova Science Publishers, New York (2009)Google Scholar
  7. 54.
    Kersten, P.: Supersymmetries and recursion operator for N = 2 supersymmetric KdV-equation. Sūrikaisekikenkyūsho Kōkyūroku 1150, 153–161 (2000)Google Scholar
  8. 59.
    Kersten, P., Krasilshchik, I., Verbovetsky, A.: (Non)local Hamiltonian and symplectic structures, recursions and hierarchies: a new approach and applications to the N = 1 supersymmetric KdV equation. J. Phys. A 37, 5003–5019 (2004)Google Scholar
  9. 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
  10. 63.
    Kersten, P., Krasilshchik, J.: Complete integrability of the coupled KdV-mKdV system. In: Morimoto, T., Sato, H., Yamaguchi, K. (eds.) Lie Groups, Geometric Structures and Differential Equations—One Hundred Years After Sophus Lie. Advanced Studies in Pure Mathematics, vol. 37, pp. 151–171. Mathematical Society of Japan, Tokyo (2002)Google Scholar
  11. 69.
    Krasilshchik, I.: A natural geometric construction underlying a class of Lax pairs. Lobachevskii J. Math. 37(1), 60–65 (2016). arXiv:1401.0612Google Scholar
  12. 71.
    Krasilshchik, I.S., Kersten, P.H.M.: Deformations and recursion operators for evolution equations. In: Prastaro, A., Rassias, T.M. (eds.) Geometry in Partial Differential Equations, pp. 114–154. World Scientific, Singapore (1994)Google Scholar
  13. 74.
    Krasilshchik, I.S., Kersten, P.H.M.: Symmetries and Recursion Operators for Classical and Supersymmetric Differential Equations. Kluwer, Dordrecht/Boston (2000)Google Scholar
  14. 76.
    Krasilshchik, I.S., Morozov, O.I., Sergyeyev, A.: Infinitely many nonlocal conservation laws for the ABC equation with A + B + C ≠ 0. Calc. Var. Partial Differ. Equ. 55(5), 1–12 (2016). arXiv:1511.09430Google Scholar
  15. 77.
    Krasilshchik, I.S., Sergyeyev, A.: Integrability of S-deformable surfaces: conservation laws, Hamiltonian structures and more. J. Geom. Phys. 97, 266–278 (2015). arXiv:1501.07171Google Scholar
  16. 104.
    Morozov, O.I., Sergyeyev, A.: The four-dimensional Martínez Alonso–Shabat equation: reductions and nonlocal symmetries. J. Geom. Phys. 85(11), 40–45 (2014)Google Scholar
  17. 113.
    Pavlov, M.V., Chang, J.H., Chen, Y.T.: Integrability of the Manakov–Santini hierarchy. arXiv:0910.2400Google Scholar
  18. 117.
    Poole, D., Hereman, W.: Symbolic computation of conservation laws for nonlinear partial differential equations in multiple space dimensions. J. Symb. Comput 46(12), 1355–1377 (2011)Google Scholar
  19. 118.
    Popovych, R.O., Samoilenko, A.M.: Local conservation laws of second-order evolution equations. J. Phys. A 41, 362002 (2008). arXiv:0806.2765Google Scholar
  20. 119.
    Popovych, R.O., Sergyeyev, A.: Conservation laws and normal forms of evolution equations. Phys. Lett. A 374, 2210–2217 (2010). arXiv:1003.1648Google Scholar
  21. 131.
    Sergyeyev, A.: New integrable (3 + 1)-dimensional systems and contact geometry. Lett. Math. Phys. (2017).
  22. 147.
    Wolf, T.: An efficiency improved program LIEPDE for determining Lie-symmetries of PDEs. In: Proceedings of Modern Group Analysis: Advanced Analytical and Computational Methods in Mathematical Physics. Kluwer (1993)Google Scholar
  23. 148.
    Wolf, T.: A comparison of four approaches to the calculation of conservation laws. Eur. J. Appl. Math. 13(2), 129–152 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 149.
    Wolf, T., Brand, A.: CRACK, user guide, examples and documentation.
  25. 150.
    Wolf, T., Brand, A.: Investigating des with crack and related programs. SIGSAM Bull. Spec. Issue 1–8 (1995)Google Scholar
  26. 152.
    Zabolotskaya, E.A., Khokhlov, R.V.: Quasi-plane waves in the nonlinear acoustics of confined beams. Sov. Phys. Acoust. 15, 35–40 (1969)Google Scholar
  27. 153.
    Zakharevich, I.: Nonlinear wave equation, nonlinear Riemann problem, and the twistor transform of Veronese webs. arXiv:math-ph/0006001Google 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