Skip to main content
SpringerLink
Log in

Amplitude modulation of water waves governed by Boussinesq’s equation

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

This paper derives the equations of amplitude modulation of shallow water waves from the scalar Boussinesq’s (BSQ) equation by using the variational-asymptotic method. Two asymptotic solutions describing the amplitude modulation of trains of solitons and of positons are obtained. The comparison with the exact solutions of BSQ equation shows quite excellent agreement.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Boussinesq, J.: Théorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond. J. Math. Pures Appl. 17, 55–108 (1872)

    MATH  Google Scholar 

  2. Whitham, G.B.: Linear and nonlinear waves. Wiley, New York (1974)

    MATH  Google Scholar 

  3. Xu, L., Auston, D.H., Hasegawa, A.: Propagation of electromagnetic solitary waves in dispersive nonlinear dielectrics. Phys. Rev. A 45, 3184–3193 (1992)

    Article  Google Scholar 

  4. Karpman, V.I.: Non-linear waves in dispersive media. Pergamon, New York (1975)

    Google Scholar 

  5. Turitsyn, S.K., Fal’kovich, G.E.: Stability of magnetoelastic solitons and self-focusing of sound in antiferromagnets, Sov. Phys. JETP 62, 146–152 (1985); Translated from Zh. Eksper. Teoret. Fiz. 89, 258–270 (1985)

  6. Yang, H.W., Yin, B.S., Shi, Y.L.: Forced dissipative Boussinesq equation for solitary waves excited by unstable topography. Nonlinear Dyn. 70, 1389–1396 (2012)

    Article  MathSciNet  Google Scholar 

  7. Yang, H.W., Wang, X.R., Yin, B.S.: A kind of new algebraic Rossby solitary waves generated by periodic external source. Nonlinear Dyn. 76, 1725–1735 (2014)

    Article  MathSciNet  Google Scholar 

  8. Zakharov, V.E., Rubenchik, A.M.: Instability of waveguides and solitons in nonlinear media. Sov. Phys. JETP 38(3), 494–500 (1974)

    Google Scholar 

  9. Ablowitz, M.J., Haberman, R.: Nonlinear evolution equations—two and three dimensions. Phys. Rev. Lett. 35(18), 1185–1188 (1975)

    Article  MathSciNet  Google Scholar 

  10. Hirota, R.: Exact N-soliton solutions of the wave equation of long waves in shallow-water and in nonlinear lattices. J. Math. Phys. 14(7), 810–814 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  11. Ablowitz, M.J., Clarkson, P.A.: Solitons, nonlinear evolution equations and inverse scattering. Cambridge University Press, Cambridge (1991)

    Book  MATH  Google Scholar 

  12. Ma, W.X., Li, C.X., He, J.: A second Wronskian formulation of the Boussinesq equation. Nonlinear Anal. 70, 4245–4258 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. Nguyen, L.T.K.: Wronskian formulation and Ansatz method for bad Boussinesq equation, Vietnam J. Math. (2014) (accepted)

  14. Whitham, G.B.: Non-linear dispersive waves. Proc. R. Soc. Lond. A 283(1393), 238–261 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  15. Kamchatnov, A.M.: Nonlinear periodic waves and their modulations. World Scientific, Singapore (2000)

    Book  MATH  Google Scholar 

  16. Gurevich, A.V., Pitaevskii, L.P.: Nonstationary structure of a collisionless shock wave. Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki 65, 590–604 (1973)

  17. El, G.A., Grimshaw, R.H.J., Pavlov, M.V.: Integrable shallow-water equations and undular bores. Stud. Appl. Math. 106(2), 157–186 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  18. El, G.A., Grimshaw, R.H.J., Kamchatnov, A.M.: Wave breaking and the generation of undular bores in an integrable shallow-water system. Stud. Appl. Math. 114(4), 395–411 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  19. Matsuno, Y., Shchesnovich, V.S., Kamchatnov, A.M., Kraenkel, R.A.: Whitham method for the Benjamin–Ono–Burgers equation and dispersive shocks. Phys. Rev. E 75(1), 016307-1–016307-5 (2007)

    Article  MathSciNet  Google Scholar 

  20. Jorge, M.C., Minzoni, A.A., Smyth, N.F.: Modulation solutions for the Benjamin–Ono equation. Phys. D: Nonlinear Phenom. 132(1), 1–18 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  21. Kamchatnov, A.M., Kraenkel, R.A., Umarov, B.A.: Asymptotic soliton train solutions of the defocusing nonlinear Schrödinger equation. Phys. Rev. E 66(3), 036609-1–036609-10 (2002)

    Article  MathSciNet  Google Scholar 

  22. Berdichevsky, V.L.: Variational–asymptotic method of constructing a theory of shells. Appl. Math. Mech. 43, 664–687 (1979)

    Article  Google Scholar 

  23. Berdichevsky, V.L.: Variational principles of continuum mechanics. Springer, Berlin (2009)

    Google Scholar 

  24. Berdichevsky, V.L.: Spatial averaging of periodic structures. Sov. Phys. Dokl. 20(5), 334–335 (1975)

    Google Scholar 

  25. Kozlov, S.M.: Averaging of random operators. Matematicheskii Sbornik 151(2), 188–202 (1979)

    Google Scholar 

  26. Jikov, V.V., Kozlov, S.M., Oleinik, O.A.: Homogenization of differential operators and integral functionals. Springer, Berlin (1994)

    Book  Google Scholar 

  27. Le, K.C.: Vibrations of shells and rods. Springer, Berlin (1999)

    Book  MATH  Google Scholar 

  28. Braides, A., Defranceschi, A.: Homogenization of multiple integrals. Clarendon Press, Oxford (1998)

    MATH  Google Scholar 

  29. Friesecke, G., James, R.D., Müller, S.: A hierarchy of plate models derived from nonlinear elasticity by gamma-convergence. Arch. Ration. Mech. Anal. 180(2), 183–236 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  30. Le, K.C., Nguyen, L.T.K.: Energy methods in dynamics. Springer, Berlin (2014)

    Book  MATH  Google Scholar 

  31. Le, K.C., Nguyen, L.T.K.: Slope modulation of waves governed by sine-Gordon equation. Commun. Nonlinear Sci. Numer. Simul. 18(7), 1563–1567 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  32. Le, K.C., Nguyen, L.T.K.: Amplitude modulation of waves governed by Korteweg–de Vries equation. Int. J. Eng. Sci. 83, 117–123 (2014)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Lehrstuhl für Mechanik - Materialtheorie, Ruhr-Universität Bochum, 44780, Bochum, Germany

    K. C. Le & L. T. K. Nguyen

Authors
  1. K. C. Le
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. L. T. K. Nguyen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to K. C. Le.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Le, K.C., Nguyen, L.T.K. Amplitude modulation of water waves governed by Boussinesq’s equation. Nonlinear Dyn 81, 659–666 (2015). https://doi.org/10.1007/s11071-015-2017-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-015-2017-1

Keywords

Search

Navigation