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Analytic solutions for the generalized complex Ginzburg–Landau equation in fiber lasers

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Abstract

Generalized complex Ginzburg–Landau equation (GCGLE) can be used to describe the nonlinear dynamic characteristics of fiber lasers and has riveted much attention of researchers in ultrafast optics. In this paper, analytic solutions of the GCGLE are obtained via the modified Hirota bilinear method. Kink waves and period waves are presented by selecting the relevant parameters. The influence of the related parameters on them is analyzed and studied. The results indicate that the desired pulses can be demonstrated by effectively controlling the dispersion and nonlinearity of fiber lasers.

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References

  1. Szmytkowski, R.: Alternative approach to the solution of the momentum-space Schrödinger equation for bound states of the N-dimensional coulomb problem. Ann. Phys. (Berlin) 524, 345–352 (2012)

    Article  MATH  Google Scholar 

  2. Dobrowolski, T.: The studies on the motion of the sine-Gordon kink on a curved surface. Ann. Phys. (Berlin) 522, 574–583 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Guo, R., Zhao, X.J.: Discrete Hirota equation: discrete Darboux transformation and new discrete soliton solutions. Nonlinear Dyn. 84, 1901–1907 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  4. Tang, B., Li, D.J.: Quantum signature of discrete breathers in a nonlinear Klein–Gordon lattice with nearest and next-nearest neighbor interactions. Commun. Nonlinear Sci. Numer. Simul. 34, 77–85 (2016)

    Article  MathSciNet  Google Scholar 

  5. Wang, L., Zhang, J.H., Wang, Z.Q., Liu, C., Li, M., Qi, F.H., Guo, R.: Breather-to-soliton transitions, nonlinear wave interactions, and modulational instability in a higher-order generalized nonlinear Schrödinger equation. Phys. Rev. E 93, 012214 (2016)

    Article  Google Scholar 

  6. Dai, C.Q., Zhang, C.Q., Fan, Y., Chen, L.: Localized modes of the (n+1)dimensional Schrödinger equation with power-law nonlinearities in PT-symmetric potentials. Commun. Nonlinear Sci. Numer. Simul. 43, 239–250 (2017)

    Article  MathSciNet  Google Scholar 

  7. Liu, W.J., Huang, L.G., Huang, P., Li, Y.Q., Lei, M.: Dark soliton control in inhomogeneous optical fibers. Appl. Math. Lett. 61, 80–87 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  8. Kong, L.Q., Dai, C.Q.: Some discussions about variable separation of nonlinear models using Riccati equation expansion method. Nonlinear Dyn. 81, 1553–1561 (2015)

  9. Dai, C.Q., Fan, Y., Zhou, G.Q., Zheng, J., Chen, L.: Vector spatiotemporal localized structures in (3+1)-dimensional strongly nonlocal nonlinear media. Nonlinear Dyn. 86, 999–1005 (2016)

    Article  MathSciNet  Google Scholar 

  10. Kong, L.Q., Liu, J., Jin, D.Q., Ding, D.J., Dai, C.Q.: Soliton dynamics in the three-spine \(\alpha \)-helical protein with inhomogeneous effect. Nonlinear Dyn. 87, 83–92 (2017)

    Article  Google Scholar 

  11. Liu, W.J., Pang, L.H., Yan, H., Ma, G.L., Lei, M., Wei, Z.Y.: High-order solitons transmission in hollow-core photonic crystal fibers. EPL 116, 64002 (2016)

    Article  Google Scholar 

  12. Denk, J., Huber, L., Reithman, E., Frey, E.: Active curved polymers form vortex patterns on membranes. Phys. Rev. Lett. 116, 178301 (2016)

    Article  Google Scholar 

  13. Park, J.: Bifurcation and stability of the generalized complex Ginzburg–Landau equation. Commun. Pure Appl. Anal. 7, 1237–1253 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Latas, S.C.V., Ferreira, M.F.S.: Impact of higher-order effects on pulsating and chaotic solitons in dissipative systems. Eur. Phys. J. Spec. Top. 223, 79–89 (2014)

    Article  Google Scholar 

  15. Wouters, M.: The stability of nonequilibrium polariton superflow in the presence of a cylindrical defect. Phys. Rev. B 84, 2461–2468 (2011)

    Article  Google Scholar 

  16. Kalashnikov, V.L., Apolonski, A.: Energy scalability of mode-locked oscillators: a completely analytical approach to analysis. Opt. Express 18, 25757–25770 (2010)

    Article  Google Scholar 

  17. Kalashnikov, V.L., Apolonski, A.: Chirped-pulse oscillators: a unified standpoint. Phys. Rev. A 79, 126–136 (2009)

    Article  Google Scholar 

  18. Zhao, X.P., Duan, N., Liu, B.: Optimal control problem of a generalized Ginzburg–Landau model equation in population problems. Math. Methods Appl. Sci. 37, 435–446 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Huang, L.G., Liu, W.J., Wong, P., Li, Y.Q., Bai, S.Y., Lei, M.: Analytic soliton solutions of cubic–quintic Ginzburg–Landau equation with variable nonlinearity and spectral filtering in fiber lasers. Ann. Phys. (Berlin) 528, 493–503 (2016)

    Article  MATH  Google Scholar 

  20. Chen, S.H., Guo, B.L.: Classical solutions of general Ginzburg–Landau equations. Acta Math. Sci. 36, 717–732 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  21. Wong, P., Pang, L.H., Huang, L.G., Li, Y.Q., Lei, M., Liu, W.J.: Dromion-like structures and stability analysis in the variable coefficients complex Ginzburg–Landau equation. Ann. Phys. 360, 341–348 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Wong, P., Pang, L.H., Wu, Y., Lei, M., Liu, W.J.: Novel asymmetric representation method for solving the higher-order Ginzburg–Landau equation. Sci. Rep. 6, 24613 (2016)

    Article  Google Scholar 

  23. Liu, W.J., Tian, B., Jiang, Y., Sun, K., Wang, P., Li, M., Qu, Q.X.: Soliton solutions and Bäcklund transformation for the complex Ginzburg–Landau equation. Appl. Math. Comput. 217, 4369–4376 (2011)

    MathSciNet  MATH  Google Scholar 

  24. Sorokin, E., Tolstik, N., Kalashnikov, V.L., Sorokina, I.T.: Chaotic chirped-pulse oscillators. Opt. Express 21, 29567–77 (2013)

    Article  Google Scholar 

  25. Latas, S.C.V., Ferreira, M.F.S., Facao, M.V.: Impact of higher-order effects on pulsating, erupting and creeping solitons. Appl. Phys. B 104, 131–137 (2011)

    Article  Google Scholar 

  26. Yomba, E., Kofane, T.C.: Exact solutions of the one-dimensional generalized modified complex Ginzburg–Landau equation. Chaos Solitons Fractals 17, 847–860 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kalashnikov, V.L., Podivilov, E., Chernykh, A., Naumov, S., Fernandez, A., Graf, R., Apolonski, A.: Approaching the microjoule frontier with femtosecond laser oscillators: theory and comparison with experiment. New J. Phys. 7, 217 (2005)

    Article  Google Scholar 

  28. Li, D.L., Dai, Z.D., Liu, X.H.: Long time behaviour for generalized complex Ginzburg–Landau equation. J. Math. Anal. Appl. 330, 934–948 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  29. Naghshband, S., Araghi, M.A.F.: Solving generalized quintic complex Ginzburg–Landau equation by homotopy analysis method. Ain Shams Eng. J. doi:10.1016/j.asej.2016.01.015

  30. Uzunov, I.M., Georgiev, Z.D.: Localized pulsating solutions of the generalized complex cubic–quintic Ginzburg–Landau equation. J. Comput. Methods Phys. 2014, 308947 (2014)

    Article  MATH  Google Scholar 

  31. Huang, L.G., Liu, W.J., Huang, P., Pan, N., Lei, M.: Soliton amplification in gain medium governed by Ginzburg–Landau equation. Nonlinear Dyn. 81, 1133–1141 (2015)

    Article  MathSciNet  Google Scholar 

  32. Kalashnikov, V.L., Sorokin, E.: Dissipative raman solitons. Opt. Express 22, 30118–30126 (2014)

    Article  Google Scholar 

  33. Guo, S.M., Mei, L.Q., He, Y.L., Ma, C.C., Sun, Y.F.: Modulation instability and dissipative ion-acoustic structures in collisional nonthermal electron–positron–ion plasma: solitary and shock waves. Plasma Sources Sci. Technol. 25, 055006 (2016)

    Article  Google Scholar 

  34. Kengne, E., Vaillancourt, R.: Modulational stability of solitary states in a lossy nonlinear electrical line. Can. J. Phys. 87, 1191–1202 (2009)

    Article  Google Scholar 

  35. Kengne, E., Lakhssassi, A., Vaillancourt, R.: Exact solutions for generalized variable-coefficients Ginzburg–Landau equation: application to Bose–Einstein condensates with multi-body interatomic interactions. J. Math. Phys. 53, 123703 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  36. Kengne, E., Vaillancourt, R.: Bose–Einstein condensates in optical lattices: the cubic–quintic nonlinear Schrödinger equation with a periodic potential. J. Phys. B 41, 205202 (2008)

  37. Kengne, E., Vaillancourt, R.: 2D Ginzburg–Landau system of complex modulation for coupled nonlinear transmission lines. J. Infrared Millim. Terahertz Waves 30, 679–699 (2009)

    Article  Google Scholar 

Download references

Acknowledgements

We thank the funding supports by the National Natural Sciences Foundation of China (Grant No. 11674036) and by the Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications, Grant No. IPOC2016ZT04).

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Authors and Affiliations

  1. State Key Laboratory of Information Photonics and Optical Communications, and School of Science, Beijing University of Posts and Telecommunications, P. O. Box 122, 100876, Beijing, China

    Wenjun Liu, Weitian Yu, Chunyu Yang, Mengli Liu, Yujia Zhang & Ming Lei

  2. Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, 100190, Beijing, China

    Wenjun Liu

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  1. Wenjun Liu
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  2. Weitian Yu
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  3. Chunyu Yang
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Correspondence to Wenjun Liu.

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Liu, W., Yu, W., Yang, C. et al. Analytic solutions for the generalized complex Ginzburg–Landau equation in fiber lasers. Nonlinear Dyn 89, 2933–2939 (2017). https://doi.org/10.1007/s11071-017-3636-5

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  • DOI: https://doi.org/10.1007/s11071-017-3636-5

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