Skip to main content
SpringerLink
Log in

Nonlinear wave solutions and their relations for the modified Benjamin–Bona–Mahony equation

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

Abstract

In this paper, we use the bifurcation method of dynamical systems to investigate the nonlinear wave solutions of the modified Benjamin–Bona–Mahony equation. These nonlinear wave solutions contain periodic wave solutions, solitary wave solutions, periodic blow-up wave solutions, kink wave solutions, unbounded wave solutions and blow-up wave solutions. Some previous results are extended.

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
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

References

  1. Benjamin, T.B., Bona, J.L., Mahony, J.J.: Model equations for long waves in nonlinear dispersive systems. Philos. Trans. R. Soc. Lond. Ser. A 272(1220), 47–78 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  2. Saut, J.C., Tzvetkov, N.: Global well-posedness for the KP-BBM equations. Appl. Math. Res. Express 2004(1), 1–16 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  3. Varlamov, V., Liu, Y.: Cauchy problem for the Ostrovsky equation. Discret. Contin. Dyn. Syst. 10(3), 731–753 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  4. Yusufoglu, E.: New solitonary solutions for the MBBM equations using Exp-function method. Phys. Lett. A 372(4), 442–446 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  5. Daghan, D., Donmez, O., Tuna, A.: Explicit solutions of the nonlinear partial differential equations. Nonlinear Anal.: Real World Appl. 11(3), 2152–2163 (2010)

  6. Abbasbandy, S., Shirzadi, A.: The first integral method for modified Benjamin–Bona–Mahony equation. Commun. Nonlinear Sci. Numer. Simul. 15(7), 1759–1764 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  7. Yusufoglu, E., Bekir, A.: The tanh and the sine–cosine methods for exact solutions of the MBBM and the Vakhnenko equations. Chaos Solitons Fract. 38(4), 1126–1133 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  8. Noor, M.A., Noor, K.I., Waheed, A., Al-Said, E.A.: Some new solitonary solutions of the modified Benjamin–Bona–Mahony equation. Comput. Math. Appl. 62(4), 2126–2131 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  9. Li, J.B., Liu, Z.R.: Smooth and non-smooth traveling waves in a nonlinearly dispersive equation. Appl. Math. Comput. 25(1), 41–56 (2000)

    MATH  Google Scholar 

  10. Wang, Q.D., Tang, M.Y.: New exact solutions for two nonlinear equations. Phys. Lett. A 372(17), 2995–3000 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  11. Tang, S.Q., Huang, W.T.: Bifurcations of travelling wave solutions for the generalized double sinh-Gordon equation. Appl. Math. Comput. 189(2), 1774–1781 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  12. Liu, Z.R., Tang, H.: Explicit periodic wave solutions and their bifurcations for generalized Camassa–Holm equation. Int. J. Bifurc. Chaos 20(8), 2507–2519 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  13. Song, M., Liu, Z.R.: Periodic wave solutions and their limits for the ZK-BBM equation. Appl. Math. Comput. 232, 9–26 (2014)

    Article  MathSciNet  Google Scholar 

  14. Song, M., Liu, Z.R.: Qualitative analysis and explicit traveling wave solutions for the Davey–Stewartson equation. Math. Methods Appl. Sci. 37, 393–401 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  15. Wen, Z.S.: Several new types of bounded wave solutions for the generalized two-component Camassa–Holm equation. Nonlinear Dyn. 77(3), 849–857 (2014)

  16. Wen, Z.S.: Bifurcation of solitons, peakons, and periodic cusp waves for \(\theta \)-equation. Nonlinear Dyn. 77(1–2), 247–253 (2014)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Shaoxing University, Shaoxing, 312000, People’s Republic of China

    Ming Song

  2. Department of Mathematics, Yuxi Normal University, Yuxi, 653100, People’s Republic of China

    Ming Song

Authors
  1. Ming Song
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ming Song.

Additional information

Research is supported by the National Natural Science Foundation of China (No. 11361069).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Song, M. Nonlinear wave solutions and their relations for the modified Benjamin–Bona–Mahony equation. Nonlinear Dyn 80, 431–446 (2015). https://doi.org/10.1007/s11071-014-1880-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-014-1880-5

Keywords

Mathematics Subject Classification

Search

Navigation