Dromion-like structures and periodic wave solutions for variable-coefficients complex cubic–quintic Ginzburg–Landau equation influenced by higher-order effects and nonlinear gain

  • Yuanyuan Yan
  • Wenjun LiuEmail author
  • Qin Zhou
  • Anjan Biswas
Original paper


In this work, the variable-coefficients complex cubic–quintic Ginzburg–Landau equation (CCQGLE) influenced by higher-order effects and nonlinear gain is considered. Based on the asymmetric method, analytic one-soliton solution for the variable-coefficients CCQGLE is constructed for the first time. In addition, with some certain conditions, the periodic wave and dromion-like structures are derived. The results obtained may be helpful in understanding the solitons amplification and solitons management in optical fiber.


Solitons Variable-coefficients Ginzburg–Landau equation Dromion-like structures Periodic wave solutions Asymmetric method 



The work of Wenjun Liu was supported by the National Natural Science Foundation of China (NSFC) (11674036; 11875008); Beijing Youth Top-Notch Talent Support Program (2017000026833ZK08); State Key Laboratory of Advanced Optical Communication Systems and Networks, Shanghai Jiao Tong University (2019GZKF03007).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Wazwaz, A.M.: A two-mode modified KdV equation with multiple soliton solutions. Appl. Math. Lett. 70, 1–6 (2017)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Wazwaz, A.M.: Abundant solutions of various physical features for the (2+1)-dimensional modified KdV-Calogero–Bogoyavlenskii–Schiff equation. Nonlinear Dyn. 89, 1727–1732 (2017)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Wazwaz, A.M., El-Tantawy, S.A.: Solving the (3+1)-dimensional KP-Boussinesq and BKP-Boussinesq equations by the simplified Hirota’s method. Nonlinear Dyn. 88, 3017–3021 (2017)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, 2277–2283 (2017)MathSciNetCrossRefGoogle Scholar
  5. 5.
    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)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Wazwaz, A.M.: A study on a two-wave mode Kadomtsev–Petviashvili equation: conditions for multiple soliton solutions to exist. Math. Methods Appl. Sci. 40, 4128–4133 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Sun, W.R., Tian, B., et al.: Rogue matter waves in a Bose–Einstein condensate with the external potential. Eur. Phys. J. D 68, 1 (2014)CrossRefGoogle Scholar
  8. 8.
    Zhang, J.: Stability of attractive Bose–Einstein condensates. J. Stat. Phys. 101, 731 (2000)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Roy, S., Bhadra, S.: Effect of two photon absorption on nonlinear pulse propagation in gain medium. Commun. Nonlinear Sci. Numer. Simul. 13(10), 2157–2166 (2008)CrossRefGoogle Scholar
  10. 10.
    Chen, Z.G., Segev, M., Christodoulides, D.N.: Optical spatial solitons: historical overview and recent advances. Rep. Prog. Phys. 75(8), 086401 (2012)CrossRefGoogle Scholar
  11. 11.
    Serkin, V.N., Hasegawa, A., Belyaeva, T.L.: Nonautonomous solitons in external potentials. Phys. Rev. Lett. 98(7), 074102 (2007)CrossRefGoogle Scholar
  12. 12.
    Serkin, V.N., Hasegawa, A., Belyaeva, T.L.: Nonautonomous matter-wave solitons near the Feshbach resonance. Phys. Rev. A 81(2), 023610 (2010)CrossRefGoogle Scholar
  13. 13.
    Konotop, V.V., Shchesnovich, V.S., Zezyulin, D.A.: Giant amplification of modes in parity-time symmetric waveguides. Phys. Lett. A 376(42–43), 2750–2753 (2012)CrossRefGoogle Scholar
  14. 14.
    Zuo, D.W., Zhang, G.F.: Exact solutions of the nonlocal Hirota equations. Appl. Math. Lett. 93, 66–71 (2019)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Yu, F.J.: Inverse scattering solutions and dynamics for a nonlocal nonlinear Gross–Pitaevskii equation with PT-symmetric external potentials. Appl. Math. Lett. 92, 108–114 (2019)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Xie, X.Y., Meng, G.Q.: Multi-dark soliton solutions for a coupled AB system in the geophysical flows. Appl. Math. Lett. 92, 201–207 (2019)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Lan, Z.Z.: Multi-soliton solutions for a (2+1)-dimensional variable-coefficient nonlinear Schrödinger equation. Appl. Math. Lett. 86, 243–248 (2018)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Li, M., Xu, T.: Dark and antidark soliton interactions in the nonlocal nonlinear Schrödinger equation with the self-induced parity-time-symmetric potential. Phys. Rev. E 91(3), 033202 (2015)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Lan, Z.Z.: Periodic, breather and rogue wave solutions for a generalized (3+1)-dimensional variable-coefficient B-type Kadomtsev–Petviashvili equation in fluid dynamics. Appl. Math. Lett. 94, 126–132 (2019)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Xu, T., Lan, S., et al.: Mixed soliton solutions of the defocusing nonlocal nonlinear Schrödinger equation. Physica D 390, 47–61 (2019)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Akhmediev, N.N., Ankiewicz, A.: Solitons, Nonlinear PuIses and Beams, pp. 267–311. Chapman and Hall, London (1997)Google Scholar
  22. 22.
    Ping, T.J., Sheng, Z.G.: Exact solitary wave solutions of higher order complex Ginzburg–Landau equation. J. Optoelectron. Laser 16(1), 120–123 (2005)Google Scholar
  23. 23.
    Mollenauer, L.F., Smith, K.: Demonstration of soliton transmission over more than 4000 km in fiber with loss periodically compensated by Raman gain. Opt. Lett. 13(8), 675–677 (1988)CrossRefGoogle Scholar
  24. 24.
    Mollenauer, L.F., Smith, K.: Experimental observation of soliton interaction over long fiber paths: discovery of a long-range interaction. Opt. Lett. 14(22), 1284–1286 (1989)CrossRefGoogle Scholar
  25. 25.
    Haus, H.A., Wong, W.S.: Solitons in optical communications. Rev. Mod. Phys. 68(2), 423 (1996)CrossRefGoogle Scholar
  26. 26.
    Newell, A.C., Whitehead, J.A.: Finite bandwidth, finite amplitude convection. Fluid Mech. 38(2), 279–303 (1969)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Liu, W.J., Pang, L.H., et al.: Dark solitons in \(\text{ WS }_{2}\) erbium-doped fiber lasers. Photonics Res. 4(3), 111–114 (2016)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Huang, L.G., Pang, L.H., et al.: Analytic soliton solutions of cubic–quintic Ginzburg–Landau equation with variable nonlinearity and spectral filtering in fiber lasers. Ann. Phys. (Berlin) 528(6), 493–503 (2016)CrossRefGoogle Scholar
  29. 29.
    Akhmediev, N.N., Ankiewicz, A.: Dissipative Solitons, pp. 305–325. Springer, Berlin (2005)CrossRefGoogle Scholar
  30. 30.
    Turitsyn, S.K., Rozanov, N.N., et al.: Dissipative solitons in fiber lasers. Physics-Uspekhi 59(7), 642–668 (2016)CrossRefGoogle Scholar
  31. 31.
    Cross, M.C., Hohenberg, P.C.: Pattern formation outside of equilibrium. Rev. Mod. Phys. 65(65), 851–1112 (1993)CrossRefGoogle Scholar
  32. 32.
    Ping, T.J., Sheng, Z.G.: Exact solitary wave solutions of higher order complex Ginzburg–Landau equation. J. Optoelectron. Laser 16(1), 120–123 (2005)Google Scholar
  33. 33.
    Uzunov, I.M., Georgiev, Z.D., Arabadzhiev, T.N.: Transitions of stationary to pulsating solutions in the complex cubic–quintic Ginzburg–Landau equation under the influence of nonlinear gain and higher-order effects. Phys. Rev. E 97(5), 052215 (2018)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Yan, Y.Y., Liu, W.J.: Stable transmission of solitons in the complex cubic–quintic Ginzburg–Landau equation with nonlinear gain and higher-order effects. Appl. Math. Lett. 98, 171–176 (2019)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Huang, S.G., Li, J., et al.: Novel spectrum properties of the periodic pi-phase-shifted fiber Bragg grating. Opt. Commun. 285(6), 1113–1117 (2012)CrossRefGoogle Scholar
  36. 36.
    Huang, L.G., Liu, W.J., et al.: Soliton amplification in gain medium governed by Ginzburg–Landau equation. Nonlinear Dyn. 81, 1133–1141 (2015)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Boiti, M., Leon, J.J.P., et al.: Scattering of localized solitons in the plane. Phys. Lett. A 132, 432 (1988)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Liu, W.J., Zhang, Y.J., et al.: Dromion-like soliton interactions for nonlinear Schrödinger equation with variable coefficients in inhomogeneous optical fibers. Nonlinear Dyn. 96(1), 729–736 (2019)CrossRefGoogle Scholar
  39. 39.
    Liu, W.J., Tian, B., Lei, M.: Dromion-like structures in the variable coefficient nonlinear Schrödinger equation. Appl. Math. Lett. 30, 28–32 (2014)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Wong, P., Pang, L.H., et al.: Novel asymmetric representation method for solving the higher-order Ginzburg–Landau equation. Sci. Rep. 6(1), 24613 (2016)CrossRefGoogle Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.State Key Laboratory of Information Photonics and Optical Communications, and School of ScienceBeijing University of Posts and TelecommunicationsBeijingPeople’s Republic of China
  2. 2.School of Electronics and Information EngineeringWuhan Donghu UniversityWuhanPeople’s Republic of China
  3. 3.Department of Physics, Chemistry and MathematicsAlabama A & M UniversityNormalUSA
  4. 4.Department of MathematicsKing Abdulaziz UniversityJeddahSaudi Arabia
  5. 5.Department of Mathematics and StatisticsTshwane University of TechnologyPretoriaSouth Africa

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