Skip to main content
SpringerLink
Log in

A multiple exp-function method for the three model equations of shallow water waves

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

Abstract

In this study, we consider three model equations of shallow water waves. Shallow water equations model the propagation of strongly nonlinear waves up to breaking and run-up in nearshore zones. We perform multiple exp-function method which is known as a generalization of Hirota’s perturbation scheme. We yield one-, two-, and three-wave solutions. The obtained solutions can be used as benchmarks for numerical solutions of the underlying equations.

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

Similar content being viewed by others

References

  1. Bekir, A.: New solitons and periodic wave solutions for some nonlinear physical models by using the sine–cosine method. Phys. Scr. 77, 045008 (2008)

    Article  MATH  Google Scholar 

  2. Bekir, A.: Application of the-expansion method for nonlinear evolution equations. Phys. Lett. A 372, 3400–3406 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Lin, J., Lou, S.Y.: Multisoliton solutions of the (3+1)-dimensional Nizhnik–Novikov–Veselov equation. Commun. Theor. Phys. 37, 265–268 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ma, W.X., Huang, T.W., Zhang, Y.: A multiple exp-function method for nonlinear differential equations and its application. Phys. Scr. 82, 065003 (2010)

    Article  MATH  Google Scholar 

  5. Ma, W.X., Abdeljabbar, A., Asaad, M.G.: Wronskian and Grammian solutions to a (3+1)-dimensional generalized KP equation. Appl. Math. Comput. 217, 10016–10023 (2011)

    MathSciNet  MATH  Google Scholar 

  6. Ma, W.X., Zhu, Z.N.: Solving the (3+1)-dimensional KP and BKP equations by the multiple exp-function algorithm. Appl. Math. Comput. 218, 11871–11879 (2012)

    MathSciNet  MATH  Google Scholar 

  7. Adem, A.R.: The generalized (1+ 1)-dimensional and (2+ 1)-dimensional Ito equations: multiple exp-function algorithm and multiple wave solutions. Comput. Math. Appl. 71, 1248–1258 (2016)

  8. Hirota, R.: The Direct Method in Soliton Theory. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  9. Hereman, W., Nuseir, A.: Symbolic methods to construct exact solutions of nonlinear partial differential equations. Math. Comput. Simul. 43, 13–27 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  10. Wazwaz, A.M.: 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)

    MathSciNet  MATH  Google Scholar 

  11. Wazwaz, A.M.: Integrable couplings of the Burgers equation and the Sharma–Tasso–Olver equation: multiple kink solutions. Rom. Rep. Phys. 65, 383–390 (2013)

    Google Scholar 

  12. Lü, X., Chen, S.T., Ma, W.X.: Constructing lump solutions to a generalized Kadomtsev–Petviashvili–Boussinesq equation. Nonlinear Dynam. 86, 523–534 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lü, X., Ma, W.X.: Study of lump dynamics based on a dimensionally reduced Hirota bilinear equation. Nonlinear Dynam. 85, 1217–1222 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  14. Wazwaz, A.M.: Two-mode fifth-order KdV equations: necessary conditions for multiple-soliton solutions to exist. Nonlinear Dynam. 87, 1685–1691 (2017)

    Article  MathSciNet  Google Scholar 

  15. Wazwaz, A.M.: Multiple soliton solutions and multiple complex soliton solutions for two distinct Boussinesq equations. Nonlinear Dynam. 85, 731–737 (2016)

    Article  MathSciNet  Google Scholar 

  16. Hietarinta, J.: A search for bilinear equations passing Hirota’s three-soliton condition. I. KdV-type bilinear equations. J. Math. Phys. 28, 1732–1742 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hietarinta, J.: A search for bilinear equations passing Hirota’s three-soliton condition. II. mKdV-type bilinear equations. J. Math. Phys. 28, 2094–2101 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  18. Wazwaz, A.M.: The Hirota’s direct method for multiple-soliton solutions for three model equations of shallow water waves. Appl. Math. Comput. 201(2008), 489–503 (1987)

    MathSciNet  MATH  Google Scholar 

  19. Wazwaz, A.M.: Solitary wave solutions of the generalized shallow water wave (GSWW) equation by Hirota’s method, tanh-coth method and Exp-function method. Appl. Math. Comput. 202, 275–286 (2008)

    MathSciNet  MATH  Google Scholar 

  20. Li, M., Guyenne, P., Li, F., Xu, L.: High order well-balanced CDG-FE methods for shallow water waves by a Green–Naghdi model. J. Comput. Phys. 257, 169–192 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. https://terrytao.wordpress.com/2011/03/13/the-shallow-water-wave-equation-and-tsunami-propagation/

  22. Demiray, H.: Weakly nonlinear waves in water of variable depth: variable-coefficient Korteweg–de Vries equation. Comput. Math. Appl. 60, 1747–1755 (2010)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

Abdullahi Rashid Adem would like to thank the Faculty Research Committee of FAST, North-West University, Mafikeng Campus, South Africa, for its financial support.

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Arts and Sciences, Uludag University, 16059, Bursa, Turkey

    Yakup Yildirim & Emrullah Yasar

  2. Department of Mathematical Sciences, Material Science Innovation and Modelling Focus Area, North-West University, Mafikeng Campus, Private Bag X 2046, Mmabatho, 2735, Republic of South Africa

    Abdullahi Rashid Adem

Authors
  1. Yakup Yildirim
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Emrullah Yasar
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Abdullahi Rashid Adem
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Abdullahi Rashid Adem.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yildirim, Y., Yasar, E. & Adem, A.R. A multiple exp-function method for the three model equations of shallow water waves. Nonlinear Dyn 89, 2291–2297 (2017). https://doi.org/10.1007/s11071-017-3588-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-017-3588-9

Keywords

Search

Navigation