Adiabatic Invariants of Second Order Korteweg-de Vries Type Equation

  • Piotr RozmejEmail author
  • Anna Karczewska
Part of the Understanding Complex Systems book series (UCS)


In this chapter we analyze the existence and forms of invariants of the extended Korteweg-de Vries equation (KdV2). This equation appears when the Euler equations for shallow water are extended to the second order, beyond Korteweg-de Vries (KdV). We show that contrary to KdV for which there is an infinite number of invariants, for KdV2 there exists only one, connected to mass (volume) conservation of the fluid. For KdV2 we found only so-called adiabatic invariants, that is, functions of the solutions which are constants neglecting terms of higher order than the order of the equation. In this chapter we present two methods for construction of such invariants. The first method, a direct one, consists in using constructions of higher KdV invariants and eliminating non-integrable terms in an approximate way. The second method introduces a near-identity transformation (NIT) which transforms KdV2 into equation (asymptotically equivalent) which is integrable. For the equation obtained by NIT, exact invariants exist, but they become approximate (adiabatic) when the inverse NIT transformation is applied and original variables are restored. Numerical tests of the exactness of adiabatic invariants for KdV2 in several cases of initial conditions are presented. These tests confirm that relative changes in these approximate invariants are small indeed. The relations of KdV invariants and KdV2 adiabatic invariants to conservation laws are discussed, as well.


Shallow water waves Nonlinear equations Invariants of KdV2 equation Adiabatic invariants 



The authors thank Prof. Eryk Infeld and Prof. George Rowlands for inspiring discussions.


  1. 1.
    Ablowitz, M.J., Clarkson, P.A.: Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge (1991)Google Scholar
  2. 2.
    Ablowitz., M.J., Segur, H.: Solitons and Inverse Scattering Transform. SIAM, Philadelphia (1981)Google Scholar
  3. 3.
    Ali, A., Kalisch, H.: On the formulation of mass, momentum and energy conservation in the KdV equation. Acta Appl. Math. 133, 113–131 (2014)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Benjamin, B.T., Olver, P.J.: Hamiltonian structure, symmetries and conservation laws for water waves. J. Fluid Mech. 125, 137–185 (1982)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Bullough, R.K., Fordy, A.P., Manakov, S.V.: Adiabatic invariants theory of near-integrable systems with damping. Phys. Lett. A 91, 98–100 (1982)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Burde, G.I., Sergyeyev, A.: Ordering of two small parameters in the shallow water wave problem. J. Phys. A: Math. Theor. 46, 075501 (2013)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Dodd, R.: On the integrability of a system of coupled KdV equations. Phys. Lett. A 89, 168–170 (1982)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    Drazin., P.G., Johnson, R.S.: Solitons: An Introduction. Cambridge University Press, Cambridge (1989)Google Scholar
  9. 9.
    Dullin, H.R., Gottwald, G.A., Holm, D.D.: An integrable shallow water equation with linear and nonlinear dispersion. Phys. Rev. Lett. 87, 194501 (2001)ADSCrossRefGoogle Scholar
  10. 10.
    Dullin, H.R., Gottwald, G.A., Holm, D.D.: Camassa-holm, Korteweg-de Vries-5 and other asymptotically equivalent equations for shallow water waves. Fluid Dyn. Res. 33, 73–95 (2003)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    Dullin, H.R., Gottwald, G.A., Holms, D.D.: On asymptotically equivalent shallow water equations. Physica D 190, 1–14 (2004)Google Scholar
  12. 12.
    Eckhaus, W., van Harten, A.: The inverse scattering method and the theory of solitons. An introduction. In: North-Holland Mathematics Studie, vol. 50. North Holland, Amsterdam (1981)Google Scholar
  13. 13.
    Fokas, A.S., Liu, Q.M.: Asymptotic integrability of water waves. Phys. Rev. Lett. 77, 2347–2351 (1996)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Gardner, C.S., Greene, J.M., Kruskal, M.D., Miura, R.M.: Method for solving the Korteweg de Vries equation. Phys. Rev. Lett. 19, 1095–1097 (1967)ADSCrossRefGoogle Scholar
  15. 15.
    Grimshaw, R.: Internal solitary waves. In: Presented at the International Conference Progress in Nonlinear Science, Held in Nizhni Novogrod, July 2001. Dedicated to the 100-th Anniversary of Alexander A. Andronov (2001)Google Scholar
  16. 16.
    Grimshaw, R., Pelinovsky., E., Talipova, T.: Modelling internal solitary waves in the costal ocean. Surv. Geophys. 28, 273–298 (2007)ADSCrossRefGoogle Scholar
  17. 17.
    He, Y.: New exact solutions for a higher order wave equation of KdV type using multiple G’/G-expansion methods. Adv. Math. Phys. 148132 (2014)Google Scholar
  18. 18.
    He, Y., Zhao, Y.M., Long, Y.: New exact solutions for a higher-order wave equation of KdV type using extended F-expansion method. Math. Prob. Eng. 128970 (2013)Google Scholar
  19. 19.
    Hiraoka, Y., Kodama, Y.: Normal forms for weakly dispersive wave equations. Lect. Notes Phys. 767, 193–196 (2009)Google Scholar
  20. 20.
    Hirota, R.: The Direct Method in Soliton Theory. Cambridge University Press, Cambridge (2004) (First published in Japanese (1992))Google Scholar
  21. 21.
    Infeld, E., Karczewska, A., Rowlands, G., Rozmej, P.: Exact cnoidal solutions of the extended KdV equation. Acta Phys. Polon. A 133, 1191–1199 (2018)CrossRefGoogle Scholar
  22. 22.
    Infeld, E., Rowlands, G.: Nonlinear Waves, Solitons and Chaos, 2nd edn. Cambridge University Press, Cambridge (2000)Google Scholar
  23. 23.
    Karczewska, A., Rozmej, P., Infeld, E.: Shallow-water soliton dynamics beyond the Korteweg de Vries equation. Phys. Rev. E 90, 012907 (2014)Google Scholar
  24. 24.
    Karczewska, A., Rozmej, P., Infeld, E.: Energy invariant for shallow-water waves and the Korteweg-de Vries equation: doubts about the invariance of energy. Phys. Rev. E 92, 053202 (2015)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    Karczewska, A., Rozmej, P., Infeld, E., Rowlands, G.: Adiabatic invariants of the extended KdV equation. Phys. Lett. A 381, 270–275 (2017)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Karczewska, A., Rozmej, P., Rutkowski, L.: A new nonlinear equation in the shallow water wave problem. Phys. Scr. 89, 054026 (2014)ADSCrossRefGoogle Scholar
  27. 27.
    Karczewska, A., Rozmej, P., Rutkowski, L.: A finite element method for extended KdV equations. Annal. UMCS Sectio AAA Phys. 70, 41–54 (2015)Google Scholar
  28. 28.
    Karczewska, A., Rozmej, P., Rutkowski, L.: Problems with energy of waves described by Korteweg-de Vries equation. Int. J. Appl. Math. Comp. Sci. 26, 555–567 (2016)CrossRefGoogle Scholar
  29. 29.
    Kodama, Y.: Normal forms for weakly dispersive wave equations. Phys. Lett. A 112, 193–196 (1985)ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    Kodama, Y.: On integrable systems with higher order corrections. Phys. Lett. A 107, 245–249 (1985)ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    Korteweg, D., de Vries, G.: On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Philos. Mag. 39, 422–443 (1895)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Luke, J.C.: A variational principle for a fluid with a free surface. J. Fluid Mech. 27, 395–397 (1967)ADSMathSciNetCrossRefGoogle Scholar
  33. 33.
    Marchant, T.R., Smyth, N.F.: The extended Korteweg-de Vries equation and the resonant flow of a fluid over topography. J. Fluid Mech. 221, 263–288 (1990)ADSMathSciNetCrossRefGoogle Scholar
  34. 34.
    Marchant, T.R., Smyth, N.F.: Soliton interaction for the extended Korteweg-de Vries equation. IMA J. Appl. Math. 56, 157–176 (1996)ADSMathSciNetCrossRefGoogle Scholar
  35. 35.
    Miura, R.M., Gardner, C.S., Kruskal, M.D.: KdV equation and generalizations II. Existence of conservation laws and constants of motion. J. Math. Phys. 9, 1204–1209 (1968)Google Scholar
  36. 36.
    Newell, A.C.: Solitons in Mathematics and Physics. Society for Industrial and Applied Mathematics, Philadelphia, PAS, New York (1985)CrossRefGoogle Scholar
  37. 37.
    Olver, P.J.: Applications of Lie groups to differential equations. Springer, New York (1993)CrossRefGoogle Scholar
  38. 38.
    Osborne, A.R.: Nonlinear ocean waves and the inverse scattering transform. Elsevier, Academic Press, Amsterdam (2010)zbMATHGoogle Scholar
  39. 39.
    Remoissenet, M.: Waves Called Solitons: Concepts and Experiments. Springer, Berlin (1999)CrossRefGoogle Scholar
  40. 40.
    Sergyeyev, A., Vitolo, R.F.: Symmetries and conservation laws for the Karczewska-Rozmej-Rutkowski-Infeld equation. Nonlinear Anal. Real World Appl. 32, 1–9 (2016)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Whitham, G.B.: Linear and Nonlinear Waves. Wiley, New York (1974)zbMATHGoogle Scholar
  42. 42.
    Yang, J.: Dynamics of embedded solitons in the extended Korteweg-de Vries equations. Stud. Appl. Math. 106, 337–365 (2001)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Zabusky, N.J.: Solitons and bound states of the time-independent Schrödinger equation. Phys. Rev. 168, 124–128 (1968)ADSCrossRefGoogle Scholar
  44. 44.
    Zhao, Y.M.: New exact solutions for a higher-order wave equation of KdV type using the multiple simplest equation method. J. Appl. Math. 848069 (2014)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of Physics and AstronomyInstitute of Physics, University of Zielona GóraZielona GóraPoland
  2. 2.Faculty of Mathematics, Computer Science and EconometricsUniversity of Zielona GóraZielona GóraPoland

Personalised recommendations