Skip to main content
SpringerLink
Log in

Combined effects of frequency and higher-order effects on soliton conversion in an erbium fiber with inhomogeneous broadening

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

Abstract

In this paper, soliton propagation in an erbium-doped fiber with inhomogeneous broadening is considered which can be described by the inhomogeneous Hirota– Maxwell–Bloch equations. The Darboux transformation method is employed to generate the soliton solution through a linear eigenvalue problem. In particular, our results demonstrate explicitly that a soliton can be converted into various nonlinear waves such as periodic wave, anti-kink soliton and flat-top soliton in the presence of higher-order effects with inhomogeneous broadening. Additionally, we found that higher-order coefficients have strong influence on the soliton transition while frequency is only responsible for phase shift. The results might be of certain value for the study of the soliton management and soliton conversion.

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

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Hasegawa, A., Tappert, F.: Transmission of stationary nonlinear optical pulses in dispersive dielectric fiber. I. Anomalous dispersion. Appl. Phys. Lett. 23, 142–144 (1973)

    Article  Google Scholar 

  2. Mollenauer, L.F., Stolen, R.H., Gordon, J.P.: Experimental observation of pico second pulse narrowing and solitons in optical fibers. Phys. Rev. Lett. 45, 1095–1098 (1980)

    Article  Google Scholar 

  3. Hasegawa, A., Kodama, Y.: Solitons in Optical Communication. Oxford University Press, New York (1995)

    MATH  Google Scholar 

  4. Hasegawa, A., Matsumoto, M.: Optical Solitons in Fibers. Springer, Berlin (2003)

    Book  Google Scholar 

  5. Serkin, V.N., Hasegawa, A.: Exactly integrable nonlinear Schrodinger equation models with varying dispersion, nonlinearity and gain: application for soliton dispersion management. IEEE J. Sel. Top. Quantum Electron. 8, 418–431 (2002)

    Article  Google Scholar 

  6. Mahalingam, A., Porsezian, K., Mani Rajan, M.S., Uthayakumar, A.: Propagation of dispersion nonlinearity managed solitons in an inhomogeneous erbium doped fiber system. J. Phys. A Math. Theor. 42, 165101 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Agrawal, G.P.: Nonlinear Fiber Optics. Academic Press, San Diego (1995)

    MATH  Google Scholar 

  8. Ankiewicz, A., Soto-Crespo, J.M., Akhmediev, N.: Rogue waves and rational solutions of the Hirota equation. Phys. Rev. E 81, 046602 (2010)

    Article  MathSciNet  Google Scholar 

  9. McCall, S.L., Hahn, E.L.: Self-induced transparency by pulsed coherent light. Phys. Rev. Lett. 18, 908 (1967)

    Article  Google Scholar 

  10. Xue, Y.S., Tian, T., Zhang, H.Q., Liu, W.J., Guo, R., Qi, F.H.: Soliton-like solutions of the coupled Hirota–Maxwell–Bloch system in optical fibers with symbolic computation. Phys. Scr. 79, 065016 (2009)

    Article  MATH  Google Scholar 

  11. Wang, L., Zhu, Y.J., Wang, Z.Q., Xu, T., Qi, F.H., Xue, Y.S.: Asymmetric Rogue waves, breather to-soliton conversion, and nonlinear wave interactions in the Hirota–Maxwell–Bloch system. J. Phys. Soc. Jpn. 85, 024001 (2016)

    Article  Google Scholar 

  12. Zakharov, V.E., Shabat, A.B.: Exact theory of two-dimensional self focusing and one dimensional self-modulation of wave in nonlinear media. Sov. Phys. JETP 34, 62 (1972)

    MathSciNet  Google Scholar 

  13. Porseian, K., Nakkeeran, K.: Optical soliton propagation in an erbium doped nonlinear light guide with higher order dispersion. Phys. Rev. Lett. 74, 2941 (1995)

    Article  Google Scholar 

  14. Glasgow, S.A., Agrotis, M.A., Ercolani, N.M.: An integrable reduction of inhomogeneously broadened optical equations. Phys. D 212, 82 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. Tian, B., Gao, Y.T., Zhu, H.W.: Variable-coefficient higher order nonlinear Schrödinger model in optical fibers: variable-coefficient bilinear form, Bäcklund transformation, brightons and symbolic computation. Phys. Lett. A 366, 223–229 (2007)

    Article  MATH  Google Scholar 

  16. Gu, C.H., Hu, H.S., Zhou, Z.X.: Darboux Transformations in Integrable Systems. Springer, New York (2005)

    Book  MATH  Google Scholar 

  17. Yu, F., Feng, L., Li, L.: Darboux transformations for super-Schrödinger equation, super-Dirac equation and their exact solutions. Nonlinear Dyn. 88, 1257–1271 (2017)

    Article  MATH  Google Scholar 

  18. Ablowitz, M.J., Kaup, D.J., Newell, A.C., Segur, H.: Nonlinear-evolution equations of physical significance. Phys. Rev. Lett. 31, 125 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  19. Mani Rajan, M.S., Mahalingam, A., Uthayakumar, A.: Nonlinear tunneling of optical soliton in 3 coupled NLS equation with symbolic computation. Ann. Phys. 346, 1 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Mani Rajan, M.S.: Dynamics of optical soliton in a tapered erbium doped fiber under periodic distributed amplification system. Nonlinear Dyn. 85, 599 (2016)

    Article  Google Scholar 

  21. Ren, Y., Yang, Z.Y., Liu, C., Yang, W.L.: Different types of nonlinear localized and periodic waves in an erbium doped fiber. Phys. Lett. A 379, 2991–2994 (2015)

    Article  Google Scholar 

  22. 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  MathSciNet  Google Scholar 

  23. Wang, L., Zhang, J.H., Liu, C., Li, M., Qi, F.H.: Breather transition dynamics, Peregrine combs and walls, and modulation instability in a variable coefficient nonlinear Schrödinger equation with higher order effects. Phys. Rev. E 93, 062217 (2016)

    Article  MathSciNet  Google Scholar 

  24. Liu, C., Yang, Z.Y., Zhao, L.C., Yang, W.L.: State transition induced by higher order effects and background frequency. Phys. Rev. E 91, 022904 (2015)

    Article  Google Scholar 

  25. Wang, L., Zhu, Y.J., Qi, F.H., Li, M., Guo, R.: Modulational instability, higher-order localized wave structures, and nonlinear wave interactions for a nonautonomous Lenells–Fokas equation in inhomogeneous fibers. Chaos 25, 063111 (2015)

    Article  MathSciNet  Google Scholar 

  26. Wang, L., Li, X., Qi, F.H., Zhang, L.L.: Breather interactions and higher-order nonautonomous rogue waves for the inhomogeneous nonlinear Schrödinger Maxwell–Bloch equations. Ann. Phys. 359, 97 (2015)

  27. Wang, L., Zhu, Y.J., Wang, Z.Q., Xu, T., Qi, F.H., Xue, Y.S.: Asymmetric Rogue waves, breather-to-soliton conversion, and nonlinear wave interactions in the Hirota–Maxwell–Bloch system. J. Phys. Soc. Jpn. 85, 024001 (2016)

    Article  Google Scholar 

  28. Wang, L., Jiang, D.Y., Qi, F.H., Shi, Y.Y., Zhao, Y.C.: Dynamics of the higher-order rogue waves for a generalized mixed nonlinear Schrödinger model. Commun. Nonlinear Sci. Numer. Simul. 42, 502 (2017)

    Article  MathSciNet  Google Scholar 

  29. Wang, X., Liu, C., Wang, L.: Darboux transformation and rogue wave solutions for the variable-coefficients coupled Hirota equations. J. Math. Anal. Appl. 449, 1534 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  30. Yomba, E., Zakeri, G.A.: Dynamics of wide and snake-like pulses in coupled Schrödinger equations with full modulated nonlinearities. Phys. Lett. A 380, 530 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  31. Tai, K., Tomita, A., Jewell, J.L., Hasegawa, A.: Generation of subpicosecond soliton like optical pulses at 0.3 THz repetition rate by induced modulation instability. Appl. Phys. Lett. 49, 236 (1986)

    Article  Google Scholar 

  32. Darvishi, M.T., Kavitha, L., Najafi, M., Senthil Kumar, V.: Elastic collision of mobile solitons of a (3+1) dimensional soliton equation. Nonlinear Dyn. 86, 765–778 (2016)

    Article  Google Scholar 

  33. Yang, J.W., Gao, Y.T., Su, C.Q., Zuo, D.W., Feng, Y.J.: Solitons and quasi periodic behaviors in an inhomogeneous optical fiber. Commun. Nonlinear Sci. Numer. Simul. 42, 477–490 (2017)

    Article  MathSciNet  Google Scholar 

  34. Chen, Y.X., Lu, X.H.: The compression and stretching of similaritons with nonlinear tunneling in optical fibers. Opt. Commun. 285, 3890 (2012)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics, University College of Engineering, Anna University, Ramanathapuram, 623513, India

    Angelin Vithya & M. S. Mani Rajan

  2. Department of EEE, University College of Engineering, Anna University, Ramanathapuram, 623513, India

    S. Arun Prakash

Authors
  1. Angelin Vithya
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. S. Mani Rajan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. Arun Prakash
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. S. Mani Rajan.

Appendix

Appendix

q(z, t):

Complex envelope of the field

p(z, t):

Polarization of the resonant medium

\(q_t \) :

Group velocity

A(z):

Averaging with respect to inhomogeneous broadening of the resonant frequency

\((\delta +\mu t)\) :

Linear inhomogeneous coefficient

\(\eta (z,t)\) :

Population inversion

\(\sigma \) :

Coefficient of second-order dispersion and self-phase modulation

\(\gamma \) :

Coefficient of third-order dispersion and self-steepening

\(\omega \) :

Frequency.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Vithya, A., Mani Rajan, M.S. & Arun Prakash, S. Combined effects of frequency and higher-order effects on soliton conversion in an erbium fiber with inhomogeneous broadening. Nonlinear Dyn 91, 687–696 (2018). https://doi.org/10.1007/s11071-017-3903-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-017-3903-5

Keywords

Search

Navigation