Journal of Engineering Mathematics

, Volume 77, Issue 1, pp 181–186 | Cite as

A note on the amplitude modulation of symmetric regularized long-wave equation with quartic nonlinearity

  • Hilmi DemirayEmail author


We study the amplitude modulation of a symmetric regularized long-wave equation with quartic nonlinearity through the use of the reductive perturbation method by introducing a new set of slow variables. The nonlinear Schrödinger (NLS) equation with seventh order nonlinearity is obtained as the evolution equation for the lowest order term in the perturbation expansion. It is also shown that the NLS equation with seventh order nonlinearity assumes an envelope type of solitary wave solution.


Nonlinear Schrödinger equation Solitary waves Symmetric regularized wave equation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Debnath L (1994) Nonlinear water waves. Academic Press, BostonzbMATHGoogle Scholar
  2. 2.
    Johnson RS (1997) A modern introduction to the mathematical theory of water waves. Cambridge University Press, CambridgezbMATHCrossRefGoogle Scholar
  3. 3.
    Colin T, Dias F, Ghidagliu (1995) On the rotational effect in the modulation of weakly nonlinear water waves over finite depth. Eur J Mech B 14: 775–793zbMATHGoogle Scholar
  4. 4.
    Dullin HR, Gottwald , Holm DD (2001) An integrable shallow water equation with linear and nonlinear dispersion. Phys Rev Lett 87: 4501–4504ADSCrossRefGoogle Scholar
  5. 5.
    Kliakhandler I, Trulsen K (2000) On weakly nonlinear modulation of waves in deep water. Phys. Fluids 12: 2432–2437MathSciNetADSCrossRefGoogle Scholar
  6. 6.
    Abourabia AM, Hassan KM, Selima ES (2010) The derivation and study of the nonlinear Schrödinger equation for long waves in shallow water using the reductive perturbation method and complex ansatz methods. Int J Nonlinear Sci 9: 430–443MathSciNetGoogle Scholar
  7. 7.
    Jukui X (2003) Modulational instability of ion acoustic waves in a plasma consisting of warm ions and non-thermal electrons. Chaos Solitons Fractals 18: 849–853ADSzbMATHCrossRefGoogle Scholar
  8. 8.
    Xue J. (2004) Cylindrical and spherical ion-acoustic solitary waves with dissipative effects. Phys. Rev. Lett. A 322: 225–230ADSzbMATHGoogle Scholar
  9. 9.
    Gill S, Kaur H, Saini NS (2006) Small amplitude electron-acoustic solitary waves in a plasma with non-thermal electrons. Chaos Solitons Fractals 30: 1020–1024ADSzbMATHCrossRefGoogle Scholar
  10. 10.
    Demiray H (2002) Contribution of higher order terms in non-linear ion-acoustic waves: strongly dispersive case. J Phys Soc Jpn 71: 1921–1930ADSCrossRefGoogle Scholar
  11. 11.
    El-Labany SK (1991) Modulation of ion-acoustic waves. Astrophys Space Sci 182: 241–247ADSzbMATHCrossRefGoogle Scholar
  12. 12.
    Ravindran R, Prasad P (1979) A mathematical analysis of nonlinear waves in a fluid-filled viscoelastic tube. Acta Mech 31: 253–280MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Erbay HA, Erbay S (1994) Nonlinear wave modulation in a fluid-filled distensible tube. Acta Mech 104: 201–214zbMATHCrossRefGoogle Scholar
  14. 14.
    Demiray H (2001) Modulation of non-linear waves in a viscous fluid contained in an elastic tube. Int J Nonlinear Mech 36: 649–661MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Demiray H (2003) On the contribution of higher order terms to solitary waves in fluid-filled elastic tubes. Int J Eng Sci 41: 1387–1403MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Demiray H (2005) Higher order approximation in reductive perturbation method: strongly dispersive waves. Commun Nonlinear Sci Numer Simul 10: 549–558MathSciNetADSzbMATHCrossRefGoogle Scholar
  17. 17.
    Benjamin T, Bonna J, Mahoney J (1978) Model equations for long waves in nonlinear dispersive systems. Phil Trans R Soc Lond Ser A 272: 47–78ADSCrossRefGoogle Scholar
  18. 18.
    Seyler CE, Fenstermacher DL (1984) A symmetrical regularized long-wave equation. Phys Fluids 27: 4–7ADSzbMATHCrossRefGoogle Scholar
  19. 19.
    Chen L (1998) Stability and instability of solitary waves for generalized symmetric regularized-long-wave equations. Physica D 118: 53–68MathSciNetADSzbMATHCrossRefGoogle Scholar
  20. 20.
    Yong C, Biao L (2004) Travelling wave solutions for generalized symmetric regularized long-wave equations with high-order nonlinear terms. Chin Phys 13: 302–306CrossRefGoogle Scholar
  21. 21.
    Demiray H (2010) Multiple time scale formalism and its applications to long water waves. Appl Math Model 34: 1187–1193MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Department of MathematicsIsik UniversityIstanbulTurkey

Personalised recommendations