Skip to main content
SpringerLink
Log in

Solving the \(\mathbf{(3+1) }\)-dimensional KP–Boussinesq and BKP–Boussinesq equations by the simplified Hirota’s method

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

Abstract

We study two (3\(+\)1)-dimensional generalized equations, namely the Kadomtsev–Petviashvili–Boussinesq equation and the B-type Kadomtsev–Petviashvili–Boussinesq equation. We use the simplified Hirota’s method to conduct this study and to find the general phase shift of these equations. We obtain one- and two-soliton solutions, for each equation, with the coefficients of the three spatial variables are left as free parameters. However, we also develop special conditions on the coefficients of the spatial variables guarantee the existence of three-soliton solutions for each of these two equation.

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

Similar content being viewed by others

References

  1. Wazwaz, A.M.: Multiple-soliton solutions for a (3 + 1)–dimensional generalized KP equation. Commun. Nonlinear. Sci. Numer. Simul. 17, 491–495 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. Wazwaz, A.M., El-Tantawy, S.: A new (3+1)-dimensional generalized Kadomtsev–Petviashvili equation. Nonlinear Dyn. 84, 1107–1112 (2016)

    Article  MathSciNet  Google Scholar 

  3. Ma, W.X., Xia, T.: Pfaffianized systems for a generalized Kadomtsev–Petviashvili equation. Phys. Scr. 87, 055003 (2013)

    Article  MATH  Google Scholar 

  4. Ma, M.A., Abdeljabbar, A.: Solving the (3+1)-dimensional generalized KP and BKP equations by the multi exp-function algorithm. Appl. Math. Comput 218, 11871–11879 (2012)

    MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  6. Biswas, A., Triki, H., Labidi, M.: Bright and dark solitons of the Rosenau-Kawahara equation with power law nonlinearity. Phys. Wave Phenom. 19(1), 24–29 (2011)

    Article  Google Scholar 

  7. Biswas, A.: Solitary wave solution for KdV equation with power-law nonlinearity and time-dependent coefficients. Nonlinear Dyn. 58(1–2), 345–348 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Biswas, A.: Solitary waves for power-law regularized long wave equation and R(m, n) equation. Nonlinear Dyn. 59(3), 423–426 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Biswas, A., Khalique, C.M.: Stationary solitons for nonlinear dispersive Schrodinger’s equation. Nonlinear Dyn. 63(4), 623–626 (2011)

    Article  Google Scholar 

  10. Kadomtsev, B.B., Petviashvili, V.I.: On the stability of solitary waves in weakly dispersive media. Sov. Phys. Dokl 15, 539–541 (1970)

    MATH  Google Scholar 

  11. El-Tantawy, S.A., Moslem, W.M., Schlickeiser,R.: Ion-acoustic dark solitons collision in an ultracold neutral plasma. Physica Scripta 90(8): 085606 (2015)

  12. El-Tantawy, S.A., Moslem, W.M.: Nonlinear structures of the Korteweg-de Vries and modified Korteweg-de Vries equations in non-Maxwellian electron-positron-ion plasma: Solitons collision and rogue waves. Phys. Plasma 21(5), 052112 (2014)

  13. 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 

  14. Baldwin, D., Hereman, W.: Symbolic software for the Painlevé test of nonlinear ordinary and partial differential equations. J. Nonlin. Math. Phys. 13(1), 90–110 (2006)

    Article  MATH  Google Scholar 

  15. Wazwaz, A.M., Xu, G.Q.: Modified Kadomtsev-Petviashvili equation in (3+1) dimensions: multiple front-wave solutions. Commun. Theor. Phys. 63, 727–730 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  16. Leblond, H., Mihalache, D.: Models of few optical cycle solitons beyond the slowly varying envelope approximation. Phys. Rep. 523, 61–126 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Leblond, H., Mihalache, D.: Few-optical-cycle solitons: modified Korteweg-de Vries sine-Gordon equation versus other non-slowly-varying-envelope-approximation models. Phys. Rev. A 79, 063835 (2009)

    Article  Google Scholar 

  18. Wazwaz, A.M.: Partial Differential Equations and Solitary Waves Theorem. Springer and HEP, Berlin (2009)

    Book  MATH  Google Scholar 

  19. Wazwaz, A.M.: \(N\)-soliton solutions for the Vakhnenko equation and its generalized forms. Phys. Scr. 82, 065006 (2010)

    Article  MATH  Google Scholar 

  20. Wazwaz, A.M.: A new generalized fifth-order nonlinear integrable equation. Phys. Scr. 83, 035003 (2011)

    Article  MATH  Google Scholar 

  21. Wazwaz, A.M.: Distinct variants of the KdV equation with compact and noncompact structures. Appl. Math. Comput 150, 365–377 (2004)

    MathSciNet  MATH  Google Scholar 

  22. Wazwaz, A.M.: New solitons and kinks solutions to the Sharma-Tasso-Olver equation. Appl. Math. Comput. 188, 1205–1213 (2007)

    MathSciNet  MATH  Google Scholar 

  23. Wazwaz, A.M.: Gaussian solitary wave solutions for nonlinear evolution equations with logarithmic nonlinearities. Nonlinear Dyn. 83(1), 591–596 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  24. Wazwaz, A.M., El-Tantawy, S.A.: A new integrable (3 + 1)-dimensional KdV-like model with its multiple-soliton solutions. Nonlinear Dyn. 83(3), 1529–1534 (2016)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Saint Xavier University, Chicago, IL, 60655, USA

    Abdul-Majid Wazwaz

  2. Department of Physics, Faculty of Science, Port Said University, Port Said, 42521, Egypt

    S. A. El-Tantawy

Authors
  1. Abdul-Majid Wazwaz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. A. El-Tantawy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Abdul-Majid Wazwaz.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wazwaz, AM., El-Tantawy, S.A. Solving the \(\mathbf{(3+1) }\)-dimensional KP–Boussinesq and BKP–Boussinesq equations by the simplified Hirota’s method. Nonlinear Dyn 88, 3017–3021 (2017). https://doi.org/10.1007/s11071-017-3429-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-017-3429-x

Keywords

Search

Navigation