Singular Perturbation of Solitons

  • Yves Pomeau
Part of the NATO ASI Series book series (NSSB, volume 284)


Long wavelength perturbation theory is at the heart of many physical approximations. It amounts to neglect derivatives and nonlinearities of higher orders, in a rational expansion. However, it may happen, that, this is formally consistent; yet the ensuing series is diverging, and so one has to investigate what really happens beyond all algebraic orders. Below, I look at a particular example of this situation, i. e. to the robustness of soliton-like solutions of the Korteweg-de Vries equation when higher order derivatives with respect to the space variable are included. This is an abridged version of a joint work with Alfred Ramani and Basile Grammaticos [1].


Travel Wave Solution Logarithmic Singularity Shallow Water Wave Stokes Line Abridge Version 
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  1. [1]
    Y. Pomeau, A. Ramani and B. Grammaticos, Physica D31, (1988), 127.MathSciNetADSGoogle Scholar
  2. [2]
    A. C. Newell “Solitons in mathematics and physics”, SIAM pub. Philadelphia (1984) and references therein.Google Scholar
  3. [3]
    M. Kruskal and H. Segur, ARAP Tech. Memo, 85–25 (1985).Google Scholar
  4. [4]
    T. Dombre, V. Hakim and Y. Pomeau, Comptes-Rendus de l’Académie des Sciences, 302 (1986) 803.MATHGoogle Scholar

Copyright information

© Plenum Press, New York 1991

Authors and Affiliations

  • Yves Pomeau
    • 1
  1. 1.Laboratoire de Physique StatistiqueParis Cedex 05France

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