Solitons and Compactons

  • Abdul-Majid Wazwaz
Part of the Nonlinear Physical Science book series (NPS)


In 1834, John Scott Russell was the first to observe the solitary waves. He observed a large protrusion of water slowly traveling on the Edinburgh-Glasgow canal without change in shape. The bulge of water, that he observed and called “great wave of translation”, was traveling along the channel of water for a long period of time wile still retaining its shape. The remarkable discovery motivated Russell to conduct physical laboratory experiments to emphasize his observance and to study these solitary waves. He empirically derived the relation
$$ c^2 = g(h + a), $$


Solitary Wave Boussinesq Equation Solitary Wave Solution Adomian Decomposition Method Exponential Tail 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    M.J. Ablowitz and P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University, Cambridge, (1991).zbMATHCrossRefGoogle Scholar
  2. 2.
    G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer, Boston, (1994).zbMATHGoogle Scholar
  3. 3.
    P.G. Drazin and R.S. Johnson, Solitons: an Introduction, Cambridge University Press, Cambridge, (1996).Google Scholar
  4. 4.
    J. Hietarinta, Introduction to the Bilinear Method, in “Integrability of Nonlinear Systems”, eds. Y. Osman-Schwarzbach, B. Grammaticos and K.M. Tamizhamani, Lecture Notes in Physics, Springer, Berlin, 638, 95–105 (2004).Google Scholar
  5. 5.
    R. Hirota, The Direct Method in Soliton Theory, Cambridge University Press, Cambridge (2004)zbMATHCrossRefGoogle Scholar
  6. 6.
    R. Hirota, Exact solutions of the Korteweg-de equation for multiple collisions of solitons, Phys. Rev. Lett., 27(18), 1192–1194, (194).CrossRefGoogle Scholar
  7. 7.
    M. Ito, An extension of nonlinear evolution equations of the KdV (mKdV) type to higher orders, J. Phys. soc. Japan, 49(2), 771–778, (1980).CrossRefMathSciNetGoogle Scholar
  8. 8.
    B.B. Kadomtsev and V.I. Petviashvili, On the stability of solitary waves in weakly dispersive media, Sov. Phys. Dokl., 15, 539–541, (1970).zbMATHGoogle Scholar
  9. 9.
    J.J.C. Nimmo and N.C. Freeman, The use of Backlund transformations in obtaining the N-soliton solutions in terms of a Wronskian, J. Phys. A: Math. General, 17, 1415–1424, (1983).CrossRefMathSciNetGoogle Scholar
  10. 10.
    P. Rosenau and J.M. Hyman, Compactons: Solitons with finite wavelenghts, Phys. Rev. Lett., 70(5), 564–567, (1993).zbMATHCrossRefGoogle Scholar
  11. 11.
    N.J. Zabusky and M.D. Kruskal, Interaction of solitons incollisionless plasma and the recurrence of initial states, Phys. Rev. Lett., 15, 240–243, (1965).CrossRefGoogle Scholar

Copyright information

© Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Abdul-Majid Wazwaz
    • 1
  1. 1.Department of MathematicsSaint Xavier UniversityChicagoUSA

Personalised recommendations