Nonlinear Dynamics

, Volume 95, Issue 3, pp 2209–2215 | Cite as

Complex simplified Hirota’s forms and Lie symmetry analysis for multiple real and complex soliton solutions of the modified KdV–Sine-Gordon equation

  • Abdul-Majid WazwazEmail author
  • Lakhveer Kaur
Original Paper


The present work consists of detailed exploration of modified KdV–Sine-Gordon equation in integrable form, owning to two-component nonlinear channel for modeling laser light propagation. For validating the behavior of this equation in the sense of integrability, we use the Painlevé test. The simplified Hirota’s technique with new complex forms is developed suitably to construct multiple-soliton solutions with complex structure for considered equation. Moreover, Lie symmetry analysis has been implemented for perceiving symmetries of MKdV–SG equation and then culminating the invariant solitary wave solutions. The new findings obviously reveal that simplified Hirota’s technique with complex structure would be highly proficient for fabricating new multiple complex soliton solutions to other nonlinear equations with integrable properties from mathematical physics and dynamical systems community.


Modified KdV–Sine-Gordon equation Simplified Hirota’s technique with complex forms Multiple-soliton solutions with complex forms Lie symmetry analysis 


Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Wazwaz, A.M.: Travelling wave solutions for the MKdV–sine-Gordon and the MKdV–sinh-Gordon equations by using a variable separated ODE method. Appl. Math. Comput. 181(2006), 1713–1719 (2006)MathSciNetzbMATHGoogle Scholar
  2. 2.
    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)CrossRefGoogle Scholar
  3. 3.
    Wazwaz, A.M.: N-soliton solutions for the integrable modified KdV–sine-Gordon equation. Phys. Scr. 89, 065805 (2014)CrossRefGoogle Scholar
  4. 4.
    Popov, S.P.: Numerical analysis of soliton solutions of the modified Korteweg–de Vries-Sine-Gordon equation. Comp. Math. Math. Phys. 55, 437–446 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Verheest, F., Olivier, C.P., Hereman, W.: Modified Korteweg–de Vries solitons at supercritical densities in two-electron temperature plasmas. J. Plasma Phys. 82, 905820208 (2016)CrossRefGoogle Scholar
  6. 6.
    Wazwaz, A.M.: Multiple real and multiple complex soliton solutions for the integrable Sine-Gordon equation. Optik 172, 622–627 (2018)CrossRefGoogle Scholar
  7. 7.
    Rethfeld, B., Ivanov, D.S., Garcia, M.E., Anisimov, S.I.: Modeling ultrafast laser ablation. J. Phys. D Appl. Phys. 50, 193001 (2017)CrossRefGoogle Scholar
  8. 8.
    Hirota, H.: The Direct Method in Soliton Theory. Cambridge University Press, Cambridge (2004)CrossRefzbMATHGoogle Scholar
  9. 9.
    Khoury, S.A.: Soliton and periodic solutions for higher order wave equations of KdV type (I). Chaos Solitons Fractals 26, 25–32 (2005)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Leblond, H., Mihalache, D.: Models of few optical cycle solitons beyond the slowly varying envelope approximation. Phys. Rep. 523, 61–126 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Mihalache, D.: Localized structures in nonlinear optical media: a selection of recent studies. Rom. Rep. Phys. 67, 1383–1400 (2013)Google Scholar
  12. 12.
    Zhou, Q., Zhu, Q.: Optical solitons in medium with parabolic law nonlinearity and higher order dispersion. Waves Random Complex Media 25, 52–59 (2014)CrossRefzbMATHGoogle Scholar
  13. 13.
    Biswas, A.: Chirp-free bright optical soliton perturbation with Fokas–Lenells equation by traveling wave hypothesis and semi-inverse variational principle. Optik 170, 431–435 (2018)CrossRefGoogle Scholar
  14. 14.
    Olver, P.J.: Applications of Lie Groups to Differential Equations. Springer, New York (1986)CrossRefzbMATHGoogle Scholar
  15. 15.
    Hereman, H., Nuseir, A.: Symbolic methods to construct exact solutions of nonlinear partial differential equations. Math. Comput. Simul. 43, 13–27 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Khalique, C.M.: Solutions and conservation laws of Benjamin–Bona–Mahony–Peregrine equation with power-law and dual power-law nonlinearities. Pramana J. Phys. 80, 413–427 (2013)CrossRefGoogle Scholar
  17. 17.
    Weiss, J.: The Painlevé property for partial differential equations II: Bäcklund transformation, Lax pairs, and the Schwarzian derivative. J. Math. Phys. 24, 1405–1413 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Wazwaz, A.M.: Partial Differential Equations and Solitary Waves Theory. Springer, Berlin (2009)CrossRefzbMATHGoogle Scholar
  19. 19.
    Wazwaz, A.M.: Abundant solutions of various physical features for the (2+1)-dimensional modified KdV-Calogero–Bogoyavlenskii–Schiff equations. Nonlinear Dyn. 89, 1727–1732 (2017)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Wazwaz, A.M.: Two-mode fifth-order KdV equations: necessary conditions for multiple soliton solutions to exist. Nonlinear Dyn. 87(3), 1685–1691 (2017)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Sun, B., Wazwaz, A.M.: General high-order breathers and rogue waves in the (3+1)-dimensional KP–Boussinesq equation. Commun. Nonlinear Sci. Numer. Simul. 64, 1–13 (2018)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Fenga, L.L., Tian, S.F., Wang, X., Zhang, T.T.: Nonlocal symmetry and consistent Riccati expansion integrability of the (1+1)-dimensional integrable nonlinear dispersive-wave system. Waves Random Complex Media 27(3), 571–586 (2017)CrossRefGoogle Scholar
  23. 23.
    Kaur, L., Gupta, R.K.: Some invariant solutions of field equations with axial symmetry for empty space containing an electrostatic field. Appl. Math. Comput. 231, 560–565 (2014)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Kaur, L., Wazwaz, A.M.: Painlevé analysis and invariant solutions of generalized fifth-order nonlinear integrable equation. Nonlinear Dyn. (2018).
  25. 25.
    Adem, K.R., Khalique, C.M.: Exact solutions and conservation laws of Zakharov–Kuznetsov modified equal width equation with power law nonlinearit. Nonlinear Anal. Real World Appl. 13, 1692–1702 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Yuan-Xi, X.: Solving mKdV–sinh-Gordon equation by a modified variable separated ordinary differential equation method. Chin. Phys. B 18(12), 5123–5132 (2009)CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of MathematicsSaint Xavier UniversityChicagoUSA
  2. 2.Department of MathematicsJaypee Institute of Information TechnologyNoidaIndia

Personalised recommendations