Skip to main content
SpringerLink
Log in

Breather-to-soliton and rogue wave-to-soliton transitions in a resonant erbium-doped fiber system with higher-order effects

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

Abstract

Under investigation in this paper is the higher-order nonlinear Schrödinger and Maxwell–Bloch system which describes the wave propagation in an erbium-doped nonlinear fiber with higher-order effects including the fourth-order dispersion and quintic non-Kerr nonlinearity. The breather and rogue wave (RW) solutions are shown that they can be converted into various soliton solutions including the multi-peak soliton, periodic wave, antidark soliton, M-shaped soliton, and W-shaped soliton. In addition, under different values of higher-order effect, the locus of the eigenvalues on the complex plane which converts breathers or RWs into solitons is calculated.

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
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  1. Agrawal, G.P.: Nonlinear Fiber Optics, 5th edn. Academic Press, London (2012)

    MATH  Google Scholar 

  2. Hasegawa, A., Tappert, F.D.: Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion. Appl. Phys. Lett. 23, 142 (1973)

    Article  Google Scholar 

  3. Porsezian, 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 

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

    Article  Google Scholar 

  5. Lamb Jr., G.L.: Elements of Soliton Theory. Wiley, New York (1980)

    MATH  Google Scholar 

  6. Doktorov, E.V., Vlasov, R.A.: Optical solitons in media with resonant and non-resonant self-focusing nonlinearities. J. Mod. Opt. 30, 223 (1983)

    Google Scholar 

  7. Maimistov, A.I., Manykin, E.A.: Propagation of ultrashort optical pulses in resonant nonlinear light guides. Sov. Phys. JETP 58, 685 (1983)

    Google Scholar 

  8. Nakazawa, M., Yamada, E., Kubota, H.: Coexistence of self-induced transparency soliton and nonlinear Schrödinger soliton. Phys. Rev. Lett. 66, 2625 (1991)

    Article  Google Scholar 

  9. Porsezian, K., Mahalingam, A., Sundaram, P.S.: Solitons in the system of coupled Hirota–Maxwell–Bloch equations. Chaos Solitons Fractals 11, 1261 (2000)

    Article  MATH  Google Scholar 

  10. Nakkeeran, K.: Optical solitons in erbium doped fibers with higher order effects. Phys. Lett. A 275, 415 (2000)

    Article  MATH  Google Scholar 

  11. Yang, B., Zhang, W.G., Zhang, H.Q., Pei, S.B.: Generalized Darboux transformation and rogue wave solutions for the higher-order dispersive nonlinear Schrödinger equation. Phys. Scr. 88, 065004 (2013)

    Article  MATH  Google Scholar 

  12. Ankiewicz, A., Wang, Y., Wabnitz, S., Akhmediev, N.: Ext-ended nonlinear Schrödinger equation with higher-order odd and even terms and its rogue wave solutions. Phys. Rev. E 89, 012907 (2014)

    Article  Google Scholar 

  13. Ankiewicz, A., Akhmediev, N.: Higher-order integrable evolution equation and its soliton solutions. Phys. Lett. A 378, 358 (2014)

    Article  MathSciNet  Google Scholar 

  14. Davydova, T.A., Zaliznyak, Y.A.: Schrödinger ordinary solitons and chirped solitons: fourth-order dispersive effects and cubic-quintic nonlinearity. Phys. D 156, 260 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  15. Li, L.J., Wu, Z.W., Wang, L.H., He, J.S.: High-order rogue waves for the Hirota equation. Ann. Phys. 334, 198 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Guo, R., Hao, H.Q.: Breathers and localized solitons for the Hirota–Maxwell–Bloch system on constant backgrounds in erbium doped fibers. Ann. Phys. 344, 10 (2014)

    Article  Google Scholar 

  17. Guo, R., Hao, H.Q.: Propagation properties of soliton solutions under the influence of higher order effects in erbium doped fibers. Commun. Nonlinear Sci. Numer. Simul. 19, 3529 (2014)

    Article  MathSciNet  Google Scholar 

  18. Guo, R., Hao, H.Q., Gu, X.S.: Modulation instability, breathers, and bound solitons in an erbium-doped fiber system with higher-order effects. Abstr. Appl. Anal. 2014, 185654 (2014)

    MathSciNet  Google Scholar 

  19. Zhang, Y., Li, C.Z., He, J.S.: Rogue waves in a resonant erbium-doped fiber system with higher-order effects. arXiv:1505.02237

  20. Wang, Q.M., Gao, Y.T., Su, C.Q., Zuo, D.W.: Solitons, breathers and rogue waves for a higher-order nonlinear Schrödinger Maxwell–Bloch system in an erbium-doped fiber system. Phys. Scr. 90, 105202 (2015)

    Article  Google Scholar 

  21. Su, C.Q., Gao, Y.T., Xue, L., Yu, X.: Solitons and rogue waves for a higher-order nonlinear Schrödinger–Maxwell–Bloch system in an erbium-doped fiber. Z. Naturforsch. A 70, 935 (2015)

    Article  Google Scholar 

  22. Zuo, D.W., Gao, Y.T., Feng, Y.J., Xue, L.: Rogue-wave interaction for a higher-order nonlinear Schrödinger–Maxwell–Bloch system in the optical-fiber communication. Nonlinear Dyn. 78, 2309 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. Li, J.T., Han, J.Z., Du, Y.D., Dai, C.Q.: Controllable behaviors of Peregrine soliton with two peaks in a birefringent fiber with higher-order effects. Nonlinear Dyn. 82, 1393 (2015)

    Article  MathSciNet  Google Scholar 

  24. Xie, X.Y., Tian, B., Sun, W.R., Sun, Y.: Rogue-wave solutions for the Kundu–Eckhaus equation with variable coefficients in an optical fiber. Nonlinear Dyn. 81, 1349 (2015)

    Article  MathSciNet  Google Scholar 

  25. Yang, Y.Q., Wang, X., Yan, Z.Y.: Optical temporal rogue waves in the generalized inhomogeneous nonlinear Schrödinger equation with varying higher-order even and odd terms. Nonlinear Dyn. 81, 833 (2015)

    Article  Google Scholar 

  26. Sun, W.R., Tian, B., Zhen, H.L., Sun, Y.: Breathers and rogue waves of the fifth-order nonlinear Schrödinger equation in the Heisenberg ferromagnetic spin chain. Nonlinear Dyn. 81, 725 (2015)

    Article  MathSciNet  Google Scholar 

  27. Meng, G.Q., Qin, J.L., Yu, G.L.: Breather and rogue wave solutions for a nonlinear Schrödinger-type system in plasmas. Nonlinear Dyn. 81, 739 (2015)

    Article  MathSciNet  Google Scholar 

  28. Chen, H.Y., Zhu, H.P.: Controllable behaviors of spatiotemporal breathers in a generalized variable-coefficient nonlinear Schrödinger model from arterial mechanics and optical fibers. Nonlinear Dyn. 81, 141 (2015)

    Article  Google Scholar 

  29. Guo, R., Liu, Y.F., Hao, H.Q., Qi, F.H.: Coherently coupled solitons, breathers and rogue waves for polarized optical waves in an isotropic medium. Nonlinear Dyn. 80, 1221 (2015)

    Article  MathSciNet  Google Scholar 

  30. Yan, Z.Y.: Two-dimensional vector rogue wave excitations and controlling parameters in the two-component Gross–Pitaevskii equations with varying potentials. Nonlinear Dyn. 79, 2515 (2015)

    Article  MathSciNet  Google Scholar 

  31. Zhu, H.P.: Spatiotemporal solitons on cnoidal wave backgrounds in three media with different distributed transverse diffraction and dispersion. Nonlinear Dyn. 76, 1651 (2014)

  32. Yu, F.J.: Matter rogue waves and management by external potentials for coupled Gross–Pitaevskii equation. Nonlinear Dyn. 80, 685 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  33. Guo, R., Hao, H.Q., Zhang, L.L.: Dynamic behaviors of the breather solutions for the AB system in fluid mechanics. Nonlinear Dyn. 74, 701 (2013)

    Article  MathSciNet  Google Scholar 

  34. Dai, C.Q., Wang, Y.Y.: Controllable combined Peregrine soliton and Kuznetsov–Ma soliton in PT-symmetric nonlinear couplers with gain and loss. Nonlinear Dyn. 80, 715 (2015)

    Article  MathSciNet  Google Scholar 

  35. Dai, C.Q., Wang, X.G., Zhou, G.Q.: Stable light-bullet solutions in the harmonic and parity-time-symmetric potentials. Phys. Rev. A 89, 013834 (2014)

    Article  Google Scholar 

  36. Dai, C.Q., Wang, Y.Y.: Superposed Akhmediev breather of the (3+1)-dimensional generalized nonlinear Schrödinger equation with external potentials. Ann. Phys. 341, 142 (2014)

    Article  Google Scholar 

  37. Dai, C.Q., Wang, Y.Y., Zhang, X.F.: Controllable Akhmediev breather and Kuznetsov–Ma soliton trains in PT-symmetric coupled waveguides. Opt. Express 22, 29862 (2014)

    Article  Google Scholar 

  38. Kedziora, D.J., Ankiewicz, A., Akhmediev, N.: Second-order nonlinear Schrödinger equation breather solutions in the degenerate and rogue wave limits. Phys. Rev. E 85, 066601 (2012)

    Article  Google Scholar 

  39. Ren, Y., Yang, Z.Y., Liu, C., Yang, W.L.: Different types of nonlinear localized and periodic waves in an erbium-doped fiber system. Phys. Lett. A 379, 45 (2015)

    Google Scholar 

  40. Chowdury, A., Ankiewicz, A., Akhmediev, N.: Moving breathers and breather-to-soliton conversions for the Hirota equation. Proc. R. Soc. A 471, 20150130 (2015)

    Article  MathSciNet  Google Scholar 

  41. Chowdury, A., Kedziora, D.J., Ankiewicz, A., Akhmediev, N.: Breather-to-soliton conversions described by the quintic equation of the nonlinear Schrödinger hierarchy. Phys. Rev. E 91, 032928 (2015)

    Article  MathSciNet  Google Scholar 

  42. Liu, C., Yang, Z.Y., Zhao, L.C., Yang, W.L.: Transition, coexistence, and interaction of vector localized waves arising from higher-order effects. Ann. Phys. 362, 130 (2015)

    Article  MathSciNet  Google Scholar 

  43. 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 

  44. Barashenkov, I.V., Smirnov, Y.S.: Existence and stability chart for the ac-driven, damped nonlinear Schrödinger solitons. Phys. Rev. E 54, 5707 (1996)

  45. Barashenkov, I.V., Zemlyanaya, E.V.: Travelling solitons in the externally driven nonlinear Schrödinger equation. J. Phys. A Math. Theor. 44, 465211 (2011)

  46. Kivshar, Y.S.: Nonlinear dynamics near the zero-dispersion point in optical fibers. Phys. Rev. A 43, 1677 (1991)

  47. Kivshar, Y.S., Afanasjev, V.V.: Dark optical solitons with reverse-sign amplitude. Phys. Rev. A 44, 1446(R) (1991)

    Article  Google Scholar 

  48. He, J.S., Xu, S.W., Ruderman, M.S., Erdelyi, R.: State transition induced by self-steepening and self phase-modulation. Chin. Phys. Lett. 31, 010502 (2014)

  49. He, J.S., Xu, S.W., Cheng, Y.: The rational solutions of the mixed nonlinear Schrödinger equation. AIP Adv. 5, 017105 (2015)

    Article  Google Scholar 

Download references

Acknowledgments

We express our sincere thanks to all the members of our discussion group for their valuable comments. This work has been supported by the National Natural Science Foundation of China under Grant No. 11305060, by the Fundamental Research Funds of the Central Universities (No. 2015ZD16), by the Innovative Talents Scheme of North China Electric Power University, and by the higher-level item cultivation project of Beijing Wuzi University (No. GJB20141001).

Author information

Authors and Affiliations

  1. Department of Mathematics and Physics, North China Electric Power University, Beijing, 102206, People’s Republic of China

    Lei Wang

  2. School of Electrical and Electronic Engineering, North China Electric Power University, Beijing, 102206, People’s Republic of China

    Shen Li

  3. School of Information, Beijing Wuzi University, Beijing, 101149, People’s Republic of China

    Feng-Hua Qi

Authors
  1. Lei Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shen Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Feng-Hua Qi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lei Wang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, L., Li, S. & Qi, FH. Breather-to-soliton and rogue wave-to-soliton transitions in a resonant erbium-doped fiber system with higher-order effects. Nonlinear Dyn 85, 389–398 (2016). https://doi.org/10.1007/s11071-016-2693-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-016-2693-5

Keywords

Search

Navigation