Applied Scientific Research

, Volume 37, Issue 1–2, pp 21–30 | Cite as

Convergence of periodic wavetrains in the limit of large wavelength

  • Jerry L. Bona


The Korteweg-de Vries equation was originally derived as a model for unidirectional propagation of water waves. This equation possesses a special class of traveling-wave solutions corresponding to surface solitary waves. It also has permanent-wave solutions which are periodic in space, the so-called cnoidal waves. A classical observation of Korteweg and de Vries was that the solitary wave is obtained as a certain limit of cnoidal wavetrains.

This result is extended here, in the context of the Korteweg-de Vries equation. It is demonstrated that a general class of solutions of the Korteweg-de Vries equation is obtained as limiting forms of periodic solutions, as the period becomes large.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Amick CJ and Toland JF Arch Rat Mech Anal (in press).Google Scholar
  2. 2.
    Benjamin TB (1974) Lectures in Applied Mathematics, vol. 15, p. 3 American Math Soc, Providence.Google Scholar
  3. 3.
    Benjamin TB, Bona JL and Mahony JJ (1972) Phil Trans Roy Soc London A 272: 47.Google Scholar
  4. 4.
    Benjamin TB, Bona JL and Bose DK (1976) Lecture notes in mathematics, vol. 503, p 207, Springer-Verlag, Berlin.Google Scholar
  5. 5.
    Bona JL and Smith R (1975) Phil Trans Roy Soc London A 278: 555.Google Scholar
  6. 6.
    Bona JL and Bose DK (1978) Fluid Mechanics Research Institute, report no. 99, University of Essex, Colchester, UK.Google Scholar
  7. 7.
    Bona JL, Bose DK and Turner REL (1981) Mathematics Research Center technical report, University of Wisconsin, Madison.Google Scholar
  8. 8.
    Broer LJF (1964) Appl Sci Res B11: 273.Google Scholar
  9. 9.
    Broer LJF (1965) Appl Sci Res B12: 113.Google Scholar
  10. 10.
    Broer LJF (1975) Appl Sci Res 31: 377.Google Scholar
  11. 11.
    Broer LJF (1975) Physica 79H: 583.Google Scholar
  12. 12.
    Broer LJF, van Groesen EWC and Timmers JMW (1976) Appl Sci Res 32: 619.Google Scholar
  13. 13.
    Deift P and Trubowitz E (1979) Comm Pure Appl Math 32: 121.Google Scholar
  14. 14.
    Kato T (1975) Lecture notes in mathematics, vol. 448, p. 25. Springer-Verlag, Berlin.Google Scholar
  15. 15.
    Korteweg DJ and de Vries G (1895) Phil Mag 39: 422.Google Scholar
  16. 16.
    Lions JL and Magenes E (1968) Problèmes aux limites non homogènes et applications, vol. 1 Dunod, Paris.Google Scholar
  17. 17.
    Lions JL (1969) Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris.Google Scholar
  18. 18.
    McKean HP and van Moerbeke P (1975) Invenciones Math 30: 217.Google Scholar
  19. 19.
    Meiss JD and Pereira NR (1978) Phys Fluids 21: 700.Google Scholar
  20. 20.
    Scott Russell J (1844) 14th meeting of the British Association for the Advancement of Science, p. 311. John Murray, London.Google Scholar
  21. 21.
    Turner REL, Annali della Scuola Normale di Pisa (in press).Google Scholar
  22. 22.
    Trubowitz E (1977) Comm Pure Appl Math 30: 321.Google Scholar
  23. 23.
    Whitham GB, (1974) Linear and nonlinear waves, John Wiley and Sons, New York.Google Scholar

Copyright information

© Martinus Nijhoff Publishers 1981

Authors and Affiliations

  • Jerry L. Bona
    • 1
  1. 1.Dept. of MathematicsThe University of ChicagoChicagoUSA

Personalised recommendations