Family of KdV-type Equations

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


In this chapter we will study a family of KdV-type equations. These equations. These equations appear in many scientific fields as will be examined for each model. This family of KdV-type equations contains the following forms:
  1. (i)
    The complex modified KdV equation [16] (CMKdV) is of the form
    $$ w_t + w_{xxx} + \alpha (|w|^2 w)_x = 0, $$
    where w is a complex valued function of the spatial coordinate x and the time t, and αis a real constant.


Soliton Solution Solitary Wave Solution Kawahara Equation Regularized Long Wave Equation Equal Width Equation 
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  1. 1.
    T.B. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid Mech., 29, 559–592, (1967).zbMATHCrossRefGoogle Scholar
  2. 2.
    R.T. Benjamin, J.L. Bona and J.J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. Roy. Soc., 272, 47–78, (1972).zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    P.G. Drazin and R.S. Johnson, Solitons: an Introduction, Cambridge University Press, Cambridge, (1996).Google Scholar
  4. 4.
    B.R. Duffy and E.J. Parkes, Travelling solitary wave solutions to a seventh-order generalized KdV equation, Phys. Lett. A, 214, 271–272, (1996).zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    W. Hereman and A. Nuseir, Symbolic methods to construct exact solutions of nonlinear partial differential equations, Math. Comput. Simulation, 43(1), 13–27, (1997).zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    R. Hirota, The Direct Method in Soliton Theory, Cambridge University Press, Cambridge (2004).zbMATHCrossRefGoogle Scholar
  7. 7.
    R. Hirota, Exact N-soliton solutions of a nonlinear wave equation, J. Math Phys., 14(7), 805–809, (1973).zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    B.B. Kadomtsev and V.I. Petviashvili, On the stability of solitary vaves in weakly dispersive media, Sov. Phys. Dokl., 15, 539–541, (1970).zbMATHGoogle Scholar
  9. 9.
    T. Kawahara, Oscillatory solitary waves in dispersive media, J. Phys. Soc. Japan, 33, 260–264, (1972).CrossRefGoogle Scholar
  10. 10.
    W-X Ma, Travelling wave solutions to a seventh order generalized KdV equation, Phys. Lett. A, 180, 221–224, (1993).CrossRefMathSciNetGoogle Scholar
  11. 11.
    Y. Matsuno, Bilinear Transformation Method, Academic Press, New York, (1984).zbMATHGoogle Scholar
  12. 12.
    Y. Matsuno, Exact multi-soliton solution of the Benjamin-Ono equation, J. Phys. A: Math. Gen., 12(4), 619–621, (1979).CrossRefGoogle Scholar
  13. 13.
    H. Ono, Algebraic solitary waves in stratified fluids, J. Phys. Soc. Japan, 39, 1082–1091, (1975).CrossRefMathSciNetGoogle Scholar
  14. 14.
    Y. Pomeau, A. Ramani and B. Grammaticos, Structural stability of the Korteweg-de Vries solitons under a singular perturbation, Physica D, 31(1), 127–134, (1988).zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    A.M. Wazwaz, New compactons, solitons and periodic solutions for nonlinear variants of the KdV and the KP equations, Chaos, Solitons and Compactons, 22(1) 249–260, (2004).zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    A.M. Wazwaz, The tanh and the sine-cosine methods for the complex modified KdV and the generalized KdV equations, Comput. Math. Applic., 49, 1101–1112, (2005).zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    A.M. Wazwaz, The tanh and the sine-cosine methods for a reliable treatment of the modified equal width equation and its variants, Commun. Nonlinear Sci. Numer. Simul., 11(2), 148–160, (2006).zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    A.M. Wazwaz, The sine-cosine and the tanh methods: reliable tools for analytic treatment of nonlinear dispersive equations, Appl. Math. Comput., 173(1), 150–164, (2006).zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    A.M. Wazwaz, The tanh-coth method for new compactons and solitons solutions for the K(n,n) and the K(n+1, n+1) equations, Appl. Math. Comput., 188, 1930–1940, (2007).zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    V.E. Zakharov and V.E. Kuznetsov, On three-dimensional solitons, Sov. Phys., 39, 285–288, (1974).Google 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

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