Multiplicative of dual-waves generated upon increasing the phase velocity parameter embedded in dual-mode Schrödinger with nonlinearity Kerr laws

  • Marwan AlquranEmail author
  • Imad Jaradat
Original Paper


In this paper, we introduced a new dual-mode nonlinear Schrödinger (DMNLS) equation with nonlinearity Kerr of types square-root law and dual-power law. The new model consists of three parameters defined as dissipative, nonlinearity and the phase velocity. Also, this model describes propagations of two simultaneously directional waves instead of single wave as in the standard Schrödinger model. We determined the necessary conditions on the dissipative nonlinearity parameters that produce soliton solutions of DMNLS. Finally, a graphical analysis regarding the effect of the phase velocity on the shapes of the obtained dual-waves is accomplished.


Dual-mode Schrödinger Square-root Kerr Dual-power Kerr Solitary wave solutions 

Mathematics Subject Classification

35C08 74J35 


Compliance with ethical standards

Conflict of interest

The authors declare that there is no conflict of interests regarding the publication of the paper.


  1. 1.
    Korsunsky, S.V.: Soliton solutions for a second-order KdV equation. Phys. Lett. A. 185, 174–176 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Lee, C.T.: Multi-Soliton Solutions of the Two-mode KdV. Ph.D. thesis Oxford University, Oxford (2007)Google Scholar
  3. 3.
    Hirota, R., Satsuma, J.: Soliton solutions of a coupled Korteweg-de Vries equation. Phys. Lett. A. 85(8–9), 407–408 (1981)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Wazwaz, A.M.: Multiple soliton solutions and other exact solutions for a two-mode KdV equation. Math. Methods Appl. Sci. 40(6), 1277–1283 (2017)MathSciNetGoogle Scholar
  5. 5.
    Xiao, Z.J., Tian, B., Zhen, H.L., Chai, J., Wu, X.Y.: Multi-soliton solutions and Bucklund transformation for a two-mode KdV equation in a fluid. Waves Random Complex Media 31(6), 1–4 (2016)zbMATHGoogle Scholar
  6. 6.
    Syam, M., Jaradat, H.M., Alquran, M.: A study on the two-mode coupled modified Korteweg-de Vries using the simplified bilinear and the trigonometric-function methods. Nonlinear Dyn 90(2), 1363–1371 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Jaradat, H.M., Syam, M., Alquran, M.: A two-mode coupled Korteweg-de Vries: multiple-soliton solutions and other exact solutions. Nonlinear Dyn 90(1), 371–377 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Alquran, M., Jaradat, H.M., Syam, M.: A modified approach for a reliable study of new nonlinear equation: two-mode Korteweg-de Vries–Burgers equation. Nonlinear Dyn. 91(3), 1619–1626 (2018)CrossRefGoogle Scholar
  9. 9.
    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, 11–301 (2010)CrossRefGoogle Scholar
  10. 10.
    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), 633–639 (1997)MathSciNetGoogle Scholar
  11. 11.
    Wazwaz A.M., Two-mode Sharma-Tasso-Olver equation and two-mode fourth-order Burgers equation: multiple kink solutions. Alex. Eng. J. (2017).
  12. 12.
    Hong, W.P., Jung, Y.D.: New non-traveling solitary wave solutions for a second-order Korteweg-de Vries equation. Z. Naturforsch. 54a, 375–378 (1999)Google Scholar
  13. 13.
    Jaradat, I., Alquran, M., Momani, S., Biswas, A.: Dark and singular optical solutions with dual-mode nonlinear Schrödinger’s equation and Kerr-law nonlinearity. Optik 172, 822–825 (2018)CrossRefGoogle Scholar
  14. 14.
    Biswas, A.: Quasi-stationary non-Kerr law optical solitons. Opt. Fiber Technol. 9(4), 224–259 (2003)CrossRefGoogle Scholar
  15. 15.
    Triki, H., Biswas, A.: Dark solitons for a generalized nonlinear Schrödinger equation with parabolic law and dual-power law nonlinearities. Math. Methods Appl. Sci. 34(8), 958–962 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Seadawy, A.R., Lu, D.: Bright and dark solitary wave soliton solutions for the generalized higher order nonlinear Schrödinger equation and its stability. Results Phys. 7, 43–48 (2017)CrossRefGoogle Scholar
  17. 17.
    Biswas, A., Asma, M., Alqahtani, R.T.: Optical soliton perturbation with Kerr law nonlinearity by Adomian decomposition method. Optik 168, 253–270 (2018)CrossRefGoogle Scholar
  18. 18.
    Biswas, A.: Theory of non-Kerr law solitons. Appl. Math. Comput 153, 369–385 (2004)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Kudryashov, N.A.: One method for finding exact solutions of nonlinear differential equations. Commun. Nonlinear Sci. Numer. Simul. 17(6), 2248–2253 (2012)Google Scholar
  20. 20.
    Wang, L., Shen, W., Meng, Y., Chen, X.: Construction of new exact solutions to time-fractional two-component evolutionary system of order \(2\) via different methods. Opt. Quantum Electron. 50, 297 (2018)CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsJordan University of Science and TechnologyIrbidJordan

Personalised recommendations