The Family of the KdV Equations

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


The ubiquitous Korteweg de-Dries (KdV) equation [14] in dimensionless variables reads
$$ u_t + auu_x + u_{xxx} = 0, $$
where subscripts denote partial derivatives. The parameter a can be scaled to any real number, where the commonly used values are a=±1 or a=±6. The KdV equation molels a variety of nonlinear phenomena, including ion acoustic waves in plasmas, and shallow water waves. The derivative u t characterizes the time evolution of the wave propagating in one direction, the nonlinear term uu x describes the steepening of the wave, and the linear term u xxx accounts for the spreading or dispersion of the wave. The KdV equation was derived by Korteweg and de Vries to describe shallow water waves of long wavelength and small amplitude. The KdV equation is a nonlinear evolution equation that models a diversity of important finite amplitude dispersive wave phenomena. It has also been used to describe a number of important physical phenomena such as acoustic waves in a harmonic crystal and ion-acoustic waves in plasmas. As stated before, this equation is the simplest nonlinear equation embodying two effects: nonlinearity represented by uu x , and linear dispersion represented by u xxx . Nonlinearity of uu x tends to localize the wave whereas dispersion spreads the wave out. The delicate balance between the weak nonlinearity of uu x and the linear dispersion of u xxx defines the formulation of solitons that consist of single humped waves. The stability of solitons is a result of the delicate equilibrium between the two effects of nonlinearity and dispersion.


Solitary Wave Soliton Solution Travel Wave Solution mKdV Equation Kink Solution 
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  1. 1.
    M.J. Ablowitz and P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge (1991).zbMATHCrossRefGoogle Scholar
  2. 2.
    M.J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, SIAM, Philadelphia (1981).zbMATHGoogle Scholar
  3. 3.
    R. Bullough and P. Caudrey, Solitons vol 17 in “Topics in Current Physics”, Springer, Berlin (1980).Google Scholar
  4. 4.
    P.G. Drazin and R.S. Johnson, Solitons: an Introduction, Cambridge University Press, Cambridge, (1996).Google Scholar
  5. 5.
    C. Gardner, J. Greene, M. Kruskal and R. Miura, Method for solving the Korteweg-deVries equation, Phys. Rev. Lett., 19, 1095–1097, (1967).zbMATHCrossRefGoogle Scholar
  6. 6.
    W. Hereman and W. Zhaung, Symbolic software for soliton theory, Acta Applicandae Mathematicae. Phys. Lett. A, 76, 95–96, (1980).CrossRefMathSciNetGoogle Scholar
  7. 7.
    W. Hereman and A. Nuseir, Symbolic methods to construct exact solutions of nonlinear partial differential equations, Math. Comput. Simulation, 43 13–27, (1997).zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    J. Hietarinta, A search for bilinear equations passing Hirota’s three-soliton condition. I. KdV-type bilinear equations, J. Math. Phys. 28(8), 1732–1742, (1987).zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    J. Hietarinta, A search for bilinear equations passing Hirota’s three-soliton condition. II. mKdV-type bilinear equations, J. Math. Phys., 28(9), 2094–2101, (1987).zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    R. Hirota, The Direct Method in Soliton Theory, Cambridge University Press, Cambridge (2004).zbMATHCrossRefGoogle Scholar
  11. 11.
    R. Hirota, Exact solutions of the Korteweg-de Vries equation for multiple collisions of solitons, Phys. Rev. Lett., 27(18), 1192–1194, (1971).zbMATHCrossRefGoogle Scholar
  12. 12.
    R. Hirota, Exact N-soliton solutions of a nonlinear wave equation, J. Math. Phys., 14(7), 805–809, (1973).zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    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
  14. 14.
    D.J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular channel, and a new type of long stationary waves, Phil. Mag. 39(5), 422–443, (1895).Google Scholar
  15. 15.
    Y. Matsuno, Bilinear Transformation Method, Academic Press, New York, (1984).zbMATHGoogle Scholar
  16. 16.
    R.M. Miura, The Korteweg-de Vries Equation: a survey of results, SIAM Rev., 18, 412–459, (1976).zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    A. Nakamura, Simple explode-decay mode solutions of a certain one-space dimensional non-linear evolution equations, J. Phys. Soc. Japan, 33(5), 1456–1458, (1972).CrossRefGoogle Scholar
  18. 18.
    P. Rosenau and J.M. Hyman, Compactons: Solitons with finite wavelengths, Phys. Rev. Lett., 70(5), 564–567, (1993).zbMATHCrossRefGoogle Scholar
  19. 19.
    A.V. Slyunaev and E.N. Pelinovski, Dynamics of large-amplitude solitons, J. Exper. Theor. Phys., 89(1), 173–181, (1999).CrossRefGoogle Scholar
  20. 20.
    M. Wadati, The exact solution of the modified Korteweg-de Vries equation, J. Phys. Soc. Japan, 32, 1681–1687, (1972).CrossRefGoogle Scholar
  21. 21.
    A.M. Wazwaz, New solitons and kink solutions for the Gardner equation, Commun. Nonlinear Sci. Numer. Simul., 12(8), 1395–1404, (2007).zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    A.M. Wazwaz, New solitons and kinks solutions to the Sharma-Tasso-Olver equation, Appl. Math. Comput., 188, 1205–1213, (2007).zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    A.M. Wazwaz, The tanh-coth method for solitons and kink solutions for nonlinear parabolic equations, Appl. Math. Comput., 188, 1467–1475, (2007).zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    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
  25. 25.
    A.M. Wazwaz, Multiple-soliton solutions for the KP equation by Hirota’s bilinear method and by the tanh-coth method, Appl. Math. Comput., 190, 633–640, (2007).zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    A.M. Wazwaz, Compact and noncompact structures for variants of the KdV equation, Appl. Math. and Comput., 132(1), 29–45, (2002).zbMATHCrossRefMathSciNetGoogle 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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