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Lobachevskii Journal of Mathematics

, Volume 39, Issue 9, pp 1305–1314 | Cite as

Symmetry Reduction and Soliton-Like Solutions for the Generalized Korteweg-De Vries Equation

  • D. Blázquez-SanzEmail author
  • J. M. Conde Martín
Part 2. Special issue “Actual Problems of Algebra and Analysis” Editors: A. M. Elizarov and E. K. Lipachev
  • 8 Downloads

Abstract

We analyze the gKdV equation, a generalized version of Korteweg-de Vries with an arbitrary function f(u). In general, for a function f(u) the Lie algebra of symmetries of gKdV is the 2-dimensional Lie algebra of translations of the plane xt. This implies the existence of plane wave solutions. Indeed, for some specific values of f(u) the equation gKdV admits a Lie algebra of symmetries of dimension grater than 2. We compute the similarity reductions corresponding to these exceptional symmetries. We prove that the gKdV equation has soliton-like solutions under some general assumptions, and we find a closed formula for the plane wave solutions, that are of hyperbolic secant type.

Keywords and phrases

Korteweg-de Vries equation Lie symmetries symmetry reduction 

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References

  1. 1.
    C. E. Kenig, G. Ponce, and V. Luis, “On the (generalized) Korteweg-de Vries equation,” Duke. Math. J. 59, 585–610 (1989).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    F. Güngör, V. Lahno, and R. Zhdanov, “Symmetry classification of KdV-type nonlinear evolution equations,” J. Math. Phys. 6, 2280–2313 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    P. Bracken, “Symmetry properties of a generalized Korteweg-de Vries equation and some explicit solutions,” Int. J. Math. Math. Sci. 13, 2159–2173 (2005).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    M. Molati and C. M. Khalique, “Group analysis of a generalized KdV equation,” Appl. Math. Inf. Sci. 8, 2845–2848 (2014).MathSciNetCrossRefGoogle Scholar
  5. 5.
    G. W. Bluman and J. D. Cole, Similarity Methods for Differential Equations, Vol. 13 of Appl. Math. Sci. (Springer, New York, Heidelberg, Berlin, 1974).CrossRefGoogle Scholar
  6. 6.
    G. W. Bluman and S. Kumei, Symmetries and Differential Equations, Vol. 81 of Appl. Math. Sci. (Springer, New York, Heidelberg, Berlin, 1989).CrossRefzbMATHGoogle Scholar
  7. 7.
    P. J. Olver, Applications of Lie Groups to Differential Equations, Vol. 107 of Graduate Texts in Mathematics (Springer, New York, Heidelberg, Berlin, 1993).CrossRefGoogle Scholar
  8. 8.
    H. Stephani, Differential Equations: Their Solutions Using Symmetries (Cambridge Univ. Press, Cambridge, 1989).zbMATHGoogle Scholar
  9. 9.
    P. G. Drazin and R. S. Johnson, Solitons: An Introduction, Cambridge Texts in Applied Mathematics (Cambridge Univ. Press, Cambridge, 1989).CrossRefGoogle Scholar
  10. 10.
    P. J. Olver and E. M. Vorob’ev, “Nonclassical and conditional symmetries,” in CRC Handbook of Lie Group Analysis ofDifferential Equations, Ed. byN. H. Ibragimov (CRC, Boca Raton, FL, 1996), Vol. 3, pp. 291–328.Google Scholar
  11. 11.
    V. I. Arnold, Ordinary Differential Equations (MIT Press, Cambridge,MA, 1978).Google Scholar
  12. 12.
    N. J. Zabusky and M. D. Kruskal, “Interaction of solitons in a collisionless plasma and the recurrence of initial states,” Phys. Rev. Lett. 15, 240–243 (1965).CrossRefzbMATHGoogle Scholar
  13. 13.
    A. L. Smyth and A. L. Worthy, “Solitary wave evolution for mKdV equations,” Wave Motion 21, 263–275 (1995).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Universidad Nacional de ColombiaSede MedellínColombia
  2. 2.Universidad San Francisco de QuitoQuitoEcuador

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