Skip to main content
SpringerLink
Log in

Two-mode fifth-order KdV equations: necessary conditions for multiple-soliton solutions to exist

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

Abstract

In this work we establish two wave modes for the integrable fifth-order Korteweg-de Vries (TfKdV) equations. We determine necessary conditions of the nonlinearity and dispersion parameters of the equation for multiple-soliton solutions to exist. We apply the simplified Hirota method to derive multiple-soliton solutions under these conditions. We also examine the dispersion relations and the phase shifts of the developed models.

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. Korsunsky, S.V.: Soliton solutions for a second-order KdV equation. Phys. Lett. A 185(1994), 174–176 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  2. Xiao, Z.-J., Tian, B., Zhen, H.-L., Chai, J., Wu, X-Yu.: Multi-soliton solutions and Bcklund transformation for a two-mode KdV equation in a fluid. Waves Random Complex Media (2016). doi:10.1080/17455030.2016.1185193

  3. Lee, C.-T., Liu, J.-L.: A Hamiltonian model and soliton phenomenon for a two-mode KdV equation. Rocky Mt. J. Math. 41(4), 1273–1289 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Lee, C.-C., Lee, C.-T., Liu, J.-L., Huang, W.-Y.: Quasi-solitons of the two-mode Korteweg-de Vries equation. Eur. Phys. J. Appl. Phys. 52, 11301 (2010)

    Article  Google Scholar 

  5. Lee, C.T., Lee, C.C.: On wave solutions of a weakly nonlinear and weakly dispersive two-mode wave system. Waves Random Complex Media 23(1), 56–76 (2013)

    Article  MathSciNet  Google Scholar 

  6. Zhu, Z., Huang, H.C., Xue, W.M.: Solitary wave solutions having two wave modes of KdV-type and KdV-Burgers-type. Chin. J. Phys. 35(6–I), 633–639 (1997)

    MathSciNet  Google Scholar 

  7. Wazwaz, A.M.: Multiple soliton solutions and other exact solutions for a two-mode KdV equation. Math. Meth. Appl. Sci. (2016). doi:10.1002/mma.4138

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

    Article  MathSciNet  MATH  Google Scholar 

  9. Biswas, A., Milovicb, D., Ranasinghec, A.: Solitary waves of Boussinesq equation in a power law media. Comm. Nonlinear Sci. Numer. Simul. 14(11), 3738–3742 (2009)

    Article  Google Scholar 

  10. 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 

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

    Article  MathSciNet  Google Scholar 

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

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Verheest, F., Olivier, C.P., Hereman, W.: Modified Korteweg-de Vries solitons at supercritical densities in two-electron temperature plasmas. J. Plasma Phys. 82(02), 905820208 (2016)

    Article  Google Scholar 

  15. Khalique, C.M.: Exact solutions and conservation laws of a coupled integrable dispersionless system. Filomat 26(5), 957–964 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Khalique, C.M.: On the solutions and conservation laws of a coupled Kadomtsev-Petviashvili equation. J. Appl. Math. 2013, 741780 (2013). doi:10.1155/2013/741780

    Article  MathSciNet  MATH  Google Scholar 

  17. Khalique, C.M.: Solutions and conservation laws of Benjamin–Bona–Mahony–Peregrine equation with power-law and dual power-law nonlinearities. Pramana 80, 413–427 (2013)

    Article  Google Scholar 

  18. Krishnan, E.V., Kumar, S., Biswas, A.: Solitons and other nonlinear waves of the Boussinesq equation. Nonlinear Dyn. 70, 1213–1221 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Leblond, M., 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 

  20. 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 

  21. Khuri, S.A.: Soliton and periodic solutions for higher order wave equations of KdV type (I). Chaos Solitons Fractals 26(1), 25–32 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  22. Khuri, S.A., Sayfy, A.: A numerical approach for solving an extended Fisher–Kolomogrov–Petrovskii–Piskunov equation. J. Comput. Appl. Math. 233(8), 2081–2089 (2010)

    Article  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  25. Wazwaz, A.M.: Gaussian solitary wave solutions for nonlinear evolution equations with logarithmic nonlinearities. Nonlinear Dyn. (2015). doi:10.1007/s11071-015-2349-x

  26. Wazwaz, A.M., El-Tantawy, S.A.: A new integrable (3 + 1)-dimensional KdV-like model with its multiple-soliton solutions. Nonlinear Dyn. (2015). doi:10.1007/s11071-015-2427-0

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

Authors
  1. Abdul-Majid Wazwaz
    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. Two-mode fifth-order KdV equations: necessary conditions for multiple-soliton solutions to exist. Nonlinear Dyn 87, 1685–1691 (2017). https://doi.org/10.1007/s11071-016-3144-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-016-3144-z

Keywords

Search

Navigation