Skip to main content
SpringerLink
Log in

Interaction and energy transition between the breather and rogue wave for a generalized nonlinear Schrödinger system with two higher-order dispersion operators in optical fibers

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

Abstract

A generalized nonlinear Schrödinger system is investigated, which can be used to describe the optical pulse propagation in inhomogeneous optical fibers with the fourth- and third-order dispersions operators. The Darboux transformation method is extended to construct a mixed breather and rogue wave solution for the system. The interaction behaviors between the breather and rogue wave are studied. As a novel result, the energy transition between the breather and rogue wave is observed. Furthermore, the impacts of the different operators on the mixed solution are analyzed.

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

Similar content being viewed by others

References

  1. Hasegawa, A., Tappert, F.: Transmission of stationary nonlinear optical physics in dispersive dielectric fibers. I: anomalous dispersion. Appl. Phys. Lett. 23, 142 (1973)

    Article  Google Scholar 

  2. Hasegawa, A., Tappert, F.: Transmission of stationary nonlinear optical physics in dispersive dielectric fibers. II: normal dispersion. Appl. Phys. Lett. 23, 171 (1973)

    Article  Google Scholar 

  3. Mitschke, F.M., Mollenauer, L.F.: Discovery of the soliton self-frequency shift. Opt. Lett. 11, 659 (1986)

    Article  Google Scholar 

  4. Porsezian, K., Daniel, M., Lakshmanan, M.: On the integrability aspects of the one dimensional classical continuum isotropic biquadratic Heisenberg spin chain. J. Math. Phys. 33, 1807 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  5. 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 

  6. Guo, R., Hao, H.Q.: Breathers and multi-soliton solutions for the higher-order generalized nonlinear Schrödinger equation. Commun. Nonlinear Sci. Numer. 18, 2426 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Wang, Y.F., Tian, B., Li, M., Wang, P., Wang, M.: Integrability and soliton-like solutions for the coupled higher-order nonlinear Schrödinger equations with variable coefficients in inhomogeneous optical fibers. Commun. Nonlinear Sci. Numer. 19, 1783 (2014)

    Article  MathSciNet  Google Scholar 

  8. 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, 97–114 (2015)

    Article  MATH  Google Scholar 

  9. Sun, W.R., Tian, B., Wang, Y.F., Zhen, H.L.: Dark single- and double-hump vector solitons of the coupled higher-order nonlinear Schrödinger equations in the birefringent or two-mode fibers. Opt. Commun. 335, 237 (2015)

    Article  Google Scholar 

  10. Arshad, M., Seadawy, A.R., Lu, D.C.: Exact bright-dark solitary wave solutions of the higher-order cubic-quintic nonlinear Schrödinger equation and its stability. Optik 138, 40 (2017)

    Article  Google Scholar 

  11. Cai, L.Y., Wang, X., Wang, L., Li, M., Liu, Y., Shi, Y.Y.: Nonautonomous multi-peak solitons and modulation instability for a variable-coefficient nonlinear Schrödinger equation with higher-order effects. Nonlinear Dyn. 90, 2221 (2017)

    Article  Google Scholar 

  12. Liu, W.J., Yang, C.Y., Liu, M.L., Yu, W.T., Zhang, Y.J., Lei, M.: Effect of high-order dispersion on three-soliton interactions for the variable-coefficients Hirota equation. Phys. Rev. E 96, 042201 (2017)

    Article  MathSciNet  Google Scholar 

  13. Sun, W.R.: Nonlinear localized wave conversions for a higher-order nonlinear Schrödinger–Maxwell–Bloch system with quintic terms in an erbium-doped fiber. Nonlinear Dyn. 89, 383 (2017)

    Article  Google Scholar 

  14. Triki, H., Azzouzi, F., Biswas, A., Moshokoa, S.P., Belic, M.: Bright optical solitons with Kerr law nonlinearity and fifth order dispersion. Optik 128, 172 (2017)

    Article  Google Scholar 

  15. Gao, X.Y.: Looking at a nonlinear inhomogeneous optical fiber through the generalized higher-order variable-coefficient Hirota equation. Appl. Math. Lett. 73, 143 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  16. Nguewawe, C.P., Yemele, D.: Properties of compact bright pulse in long optical waveguide: higher-order effects. Optik 130, 1 (2017)

    Article  Google Scholar 

  17. Vithya, A., Rajan, M.S.M., Prakash, S.A.: Combined effects of frequency and higher-order effects on soliton conversion in an erbium fiber with inhomogeneous broadening. Nonlinear Dyn. 91, 687 (2018)

    Article  Google Scholar 

  18. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. Liu, W., Qiu, D.Q., He, J.S.: Localized properties of rogue wave for a higher-order nonlinear Schrödinger equation. Commun. Theor. Phys. 63, 525 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Su, C.Q., Qin, N., Li, J.G.: Conservation laws, nonautonomous breathers and rogue waves for a higher-order nonlinear Schrödinger equation in the inhomogeneous optical fiber. Superlattices Microstruct. 100, 381 (2016)

    Article  Google Scholar 

  22. Chettouh, S., Triki, H., El-Akrmi, A., Zhou, Q., Moshokoa, S.P., Ullah, M.Z., Biswas, A., Belic, M.: Dipole solitons in an extended nonlinear Schrödinger’s equation with higher-order even and odd terms. Optik 145, 644 (2017)

    Article  Google Scholar 

  23. Su, D., Yong, X.L., Tian, Y.J., Tian, J.: Breather and rogue wave solutions of an extended nonlinear Schrödinger equation with higher-order odd and even terms. Mod. Phys. Lett. B 32, 1850309 (2018)

    Article  Google Scholar 

  24. Cao, B., Zhang, H.: Conversions and interactions of the nonlinear waves in a generalized higher-order nonlinear Schrödinger equation. Optik 158, 112 (2018)

    Article  Google Scholar 

  25. Guan, W.Y., Li, B.Q.: New observation on the breather for a generalized nonlinear Schrödinger system with two higher-order dispersion operators in inhomogeneous optical fiber. Optik 181, 853 (2019)

    Article  Google Scholar 

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

    Article  Google Scholar 

  27. 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  MATH  Google Scholar 

  28. Chai, J., Tian, B., Zhen, H.L., Sun, W.R.: Conservation laws, bilinear forms and solitons for a fifth-order nonlinear Schrödinger equation for the attosecond pulses in an optical fiber. Ann. Phys. 359, 371 (2015)

    Article  MATH  Google Scholar 

  29. Yomba, E., Zakeri, G.A.: N-soliton interactions in an extended Schrödinger equation with higher order of nonlinearities. Physica B 483, 26 (2016)

    Article  Google Scholar 

  30. Chowdury, A., Krolikowski, W., Akhmediev, N.: Breather solutions of a fourth-order nonlinear Schrödinger equation in the degenerate, soliton, and rogue wave limits. Phys. Rev. E 96, 042209 (2017)

    Article  MathSciNet  Google Scholar 

  31. Sun, Y., Tian, B., Wu, X.Y., Liu, L., Yuan, Y.Q.: Dark solitons for a variable-coefficient higher-order nonlinear Schrödinger equation in the inhomogeneous optical fiber. Mod. Phys. Lett. B 31, 1750065 (2017)

    Article  Google Scholar 

  32. Bendahmane, I., Triki, H., Biswas, A., Alshomrani, A.S., Zhou, Q., Moshokoa, S.P., Belic, M.: Bright, dark and W-shaped solitons with extended nonlinear Schrödinger’s equation for odd and even higher-order terms. Superlattices Microstruct. 114, 53 (2018)

    Article  Google Scholar 

  33. Triki, H., Choudhuri, A., Porsezian, K., Dinda, P.: Dark solitons in an extended nonlinear Schrödinger equation with higher-order odd and even terms. Optik 164, 661 (2018)

    Article  Google Scholar 

  34. Triki, H., Biswas, A., Moshokoa, S.P., Belic, M.: Optical solitons and conservation laws with quadratic-cubic nonlinearity. Optik 128, 63 (2017)

    Article  Google Scholar 

  35. 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 (2017)

    Article  MathSciNet  Google Scholar 

  36. Islam, W., Younis, M., Rizvi, S.T.R.: Optical solitons with time fractional nonlinear Schrödinger equation and competing weakly nonlocal nonlinearity. Optik 130, 526 (2017)

    Article  Google Scholar 

  37. Ma, Y.L., Li, B.Q., Fu, Y.Y.: A series of the solutions for the Heisenberg ferromagnetic spin chain equation. Math. Methods Appl. Sci. 41, 3316 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  38. Li, B.Q., Ma, Y.L.: The multiple-lump waves for a (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equation arising from incompressible fluid. Comput. Math. Appl. 76, 204 (2018)

    Article  MathSciNet  Google Scholar 

  39. Li, B.Q., Ma, Y.L., Mo, L.P., Fu, Y.Y.: The N-loop soliton solutions for (2+1)-dimensional Vakhnenko equation. Comput. Math. Appl. 74, 504 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  40. Si, H.L., Li, B.Q.: Two types of soliton twining behaviors for the Kraenkel–Manna–Merle system in saturated ferromagnetic materials. Optik 166, 49 (2018)

    Article  Google Scholar 

  41. Ankiewicz, A., Akhmediev, A.: Rogue wave-type solutions of the mKdV equation and their relation to known NLSE rogue wave solutions. Nonlinear Dyn. 91, 1931 (2018)

    Article  Google Scholar 

  42. Li, B.Q., Ma, Y.L.: Solitons resonant behavior for a waveguide directional coupler system in optical fibers. Opt. Quant. Electron. 50, 270 (2018)

    Article  Google Scholar 

  43. Li, B.Q., Ma, Y.L.: Loop-like periodic waves and solitons to the Kraenkel–Manna–Merle system in ferrites. J. Electromagnet. Waves Appl. 32, 1275 (2018)

    Article  Google Scholar 

  44. Ma, Y.L., Li, B.Q.: Analytic rogue wave solutions for a generalized fourth-order Boussinesq equation in fluid mechanics. Math. Methods Appl. Sci. 42, 39 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  45. Wazwaz, A.M.: A new integrable (2+1)-dimensional generalized breaking soliton equation: N-soliton solutions and traveling wave solutions. Commun. Theor. Phys. 66, 385 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  46. Yan, X.W., Tian, S.F., Dong, M.J., Zhou, L., Zhang, T.T.: Characteristics of solitary wave, homoclinic breather wave and rogue wave solutions in a (2+1)-dimensional generalized breaking soliton equation. Comput. Math. Appl. 76, 179 (2018)

    Article  MathSciNet  Google Scholar 

  47. Li, B.Q., Ma, Y.L.: Characteristics of rogue waves for a (2+1)-dimensional Heisenberg ferromagnetic spin chain system. J. Magn. Magn. Mater. 474, 537 (2019)

    Article  Google Scholar 

  48. Ohta, Y., Yang, J.K.: General high-order rogue waves and their dynamics in the nonlinear Schrödinger equation. Proc. R. Soc. A Math. Phys. 468, 1716 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  49. Guo, B.L., Ling, L.M., Liu, Q.P.: Nonlinear Schrödinger equation generalized Darboux transformation and rogue wave solutions. Phys. Rev. E 85, 026607 (2012)

    Article  Google Scholar 

  50. Onorato, M., Residori, S., Bortolozzo, U., Montina, A., Arecchi, F.T.: Rogue waves and their generating mechanisms in different physical contexts. Phys. Rep. 528, 47 (2013)

    Article  MathSciNet  Google Scholar 

  51. Ling, L.M., Zhao, L.C., Guo, B.L.: Darboux transformation and classification of solution for mixed coupled nonlinear Schrödinger equations. Commun. Nonlinear Sci. Numer. 32, 285 (2016)

    Article  MathSciNet  Google Scholar 

  52. Li, B.Q., Ma, Y.L.: Lax pair, Darboux transformation and Nth-order rogue wave solutions for a (2+1)-dimensional Heisenberg ferromagnetic spin chain equation. Comput. Math. Appl. 77, 514 (2019)

    Article  MathSciNet  Google Scholar 

  53. Ma, Y.L., Li, B.Q.: Interactions between rogue wave and soliton for a (2+1)-dimensional generalized breaking soliton system: hidden rogue wave and hidden soliton. Comput. Math. Appl. (2019). https://doi.org/10.1016/j.camwa.2019.03.002

    Google Scholar 

  54. Guo, B.L., Ling, L.M.: Rogue wave, breathers and bright–dark–rogue solutions for the coupled Schrodinger equations. Chin. Phys. Lett. 28, 110202 (2011)

    Article  Google Scholar 

  55. Harun-Or-Roshid, Ma, W.X.: Dynamics of mixed lump-solitary waves of an extended (2+1)-dimensional shallow water wave model. Phys. Lett. A 382, 3262 (2018)

Download references

Acknowledgements

This work was supported partially by the National Natural Science Foundation of China under Grant No. 11571023.

Author information

Authors and Affiliations

  1. School of Science, Beijing Technology and Business University, Beijing, 100048, China

    Yu-Lan Ma

Authors
  1. Yu-Lan Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yu-Lan Ma.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest to this work. There is no professional or other personal interest of any nature or kind in any product that could be construed as influencing the position presented in the manuscript entitled.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ma, YL. Interaction and energy transition between the breather and rogue wave for a generalized nonlinear Schrödinger system with two higher-order dispersion operators in optical fibers. Nonlinear Dyn 97, 95–105 (2019). https://doi.org/10.1007/s11071-019-04956-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-019-04956-0

Keywords

Search

Navigation