Skip to main content
Log in

Soliton interaction control through dispersion and nonlinear effects for the fifth-order nonlinear Schrödinger equation

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

Abstract

Optical fiber communication has developed rapidly because of the needs of the information age. Here, the variable coefficients fifth-order nonlinear Schrödinger equation (NLS), which can be used to describe the transmission of femtosecond pulse in the optical fiber, is studied. By virtue of the Hirota method, we get the one-soliton and two-soliton solutions. Interactions between solitons are presented, and the soliton stability is discussed through adjusting the values of dispersion and nonlinear effects. Results may potentially be useful for optical communications such as all-optical switches or the study of soliton control.

This is a preview of subscription content, log in via an institution to check access.

Access this article

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

Availability of data and material

The authors declare that all data generated or analyzed during this study are included in this article.

References

  1. 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(1), 012907 (2014)

    Article  Google Scholar 

  2. Wazwaz, A.M.: A two-mode modified KdV equation with multiple soliton solutions. Appl. Math. Lett. 70, 1–6 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  3. 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)

    Article  MathSciNet  Google Scholar 

  4. Wazwaz, A.M., EI-Tantawy, S.A.: Solving the \((3+1)\)-dimensional KP-Boussinesq and BKP-Boussinnesq equation by the simplified Hirota’s method. Nonlinear Dyn. 88, 3017–3021 (2017)

  5. Wazwaz, A.M.: Multiple soliton solutions and other exact solutions for a two-mode KdV equation. Math. Methods Appl. Sci. 40, 2277–2283 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  6. Wazwaz, A.M., EI-Tantawy, S.A. : A new integrable \((3+1)\)-dimensional KdV-like model with its multiple-soliton solutions. Nonlinear Dyn. 83(3), 1529–1534 (2016)

  7. 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)

    Article  MathSciNet  MATH  Google Scholar 

  8. Zhang, N., Xia, T.C., Fan, E.G.: A Riemann-Hilbert approach to the Chen–Lee–Liu equation on the half line. Acta Math. Appl. Sin. 34(3), 493–515 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  9. Zhang, N., Xia, T.C., Jin, Q.Y.: N-Fold Darboux transformation of the discrete Ragnisco–Tu system. Adv. Differ. Equ. 2018, 302 (2018)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  11. Liu, W.J., Zhang, Y.J., Pang, L.H., Yan, H., Ma, G.L., Lei, M.: Study on the control technology of optical solitons in optical fibers. Nonlinear Dyn. 86, 1069–1073 (2016)

    Article  Google Scholar 

  12. Hasegawa, A., Tappert, F.: Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion. Appl. Phys. Lett. 23, 142–144 (2003)

    Article  Google Scholar 

  13. Hirota, R.: The Direct Method in Soliton Theory. Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  14. Kivshar, Y.S., Haelterman, M., Emplit, P., Hamaide, J.P.: Gordon–Haus effect on dark solitons. Opt. Lett. 19, 19–21 (1994)

    Article  Google Scholar 

  15. Liu, M.L., Liu, W.J., Pang, L.H., Teng, H., Fang, S.B., Wei, Z.Y.: Ultrashort pulse generation in mode-locked erbium-doped fiber lasers with tungsten disulfide saturable absorber. Opt. Commun. 406, 72–75 (2018)

    Article  Google Scholar 

  16. Liu, W.J., Liu, M.L., OuYang, Y.Y., Hou, H.R., Ma, G.L., Lei, M., Wei, Z.Y.: Tungsten diselenide for mode-locked erbium-doped fiber lasers with short pulse duration. Nanotechnology 29, 174002 (2018)

    Article  Google Scholar 

  17. Liu, W.J., Liu, M.L., Lei, M., Fang, S.B., Wei, Z.Y.: Titanium selenide saturable absorber mirror for passive Q-switched Er-doped fiber laser. IEEE J. Quantum. Elect. 24, 0901005 (2017)

    Google Scholar 

  18. Liu, W.J., Pang, L.H., Han, H.N., Tian, W.L., Chen, H., Lei, M., Yan, P.G., Wei, Z.Y.: Generation of dark solitons in erbium-doped fiber lasers based Sb\(_2\)Te\(_3\) saturable absorbers. Opt. Express 23, 26023–26031 (2015)

    Article  Google Scholar 

  19. Kumar, S., Niwas, M., Wazwaz, A.M.: Lie symmetry analysis, exact analytical solutions and dynamics of solitons for (2+1)-dimensional NNV equations. Phys. Scr. 95(9), 095204 (2020)

    Article  Google Scholar 

  20. Kumar, S., Kumar, D., Wazwaz, A.M.: Lie symmetries, optimal system, group-invariant solutions and dynamical behaviors of solitary wave solutions for a \((3+1)\)-dimensional KdV-type equation. Eur. Phys. J. Plus 136, 531 (2021)

    Article  Google Scholar 

  21. Kumar, S., Kumar, A.: Lie symmetry reductions and group invariant solutions of \((2+1)\)-dimensional modified Veronese web equation. Nonlinear Dyn. 98, 1891–1903 (2019)

    Article  MATH  Google Scholar 

  22. Kumar, S., Kumar, D., Wazwaz, A.M.: Group invariant solutions of \((3+1)\)-dimensional generalized B-type Kadomstsev Petviashvili equation using optimal system of Lie subalgebra. Phys. Scr. 94(6), 065204 (2019)

    Article  Google Scholar 

  23. Kumar, S., Almusawa, H., Hamid, I., Abdou, M.A.: Abundant closed-form solutions and solitonic structures to an integrable fifth-order generalized nonlinear evolution equation in plasma physics. Results Phys. 26, 104453 (2021)

    Article  Google Scholar 

  24. Kumar, S., Kumar, A.: Abundant closed-form wave solutions and dynamical structures of soliton solutions to the \((3+1)\)-dimensional BLMP equation in mathematical physics. J. Ocean Eng. Sci. (In Press). https://doi.org/10.1016/j.joes.2021.08.001

  25. Ankiewicz, A., Akhmediev, N.: Higher-order integrable evolution equation and its soliton solutions. Phys. Lett. A 378(4), 358–361 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  26. Chen, C.J., Wang, P.K., Menyuk, C.R.: Soliton switch using birefringent optical fibers. Opt. Lett. 15(9), 477 (1990)

    Article  Google Scholar 

  27. Dai, C.Q., Zhou, G.Q., Chen, R.P., Lai, X.J., Zheng, J.: Vector multipole and vortex solitons in two-dimensional Kerr media. Nonlinear Dyn. 88, 2629–2635 (2017)

    Article  MathSciNet  Google Scholar 

  28. Abdullaev, F.K., Garnier, J.: Dynamical stabilization of solitons in cubic-quintic nonlinear Schrödinger model. Phys. Rev. E 72, 035603R (2005)

    Article  MathSciNet  Google Scholar 

  29. Mollenauer, L.F., Neubelt, M.J., Evangelides, S.G., Gordon, J.P., Simpson, J.R., Cohen, L.G.: Experimental study of soliton transmission over more than 10,000 km in dispersion-shifted fiber. Opt. Lett. 15, 1203–1205 (1990)

    Article  Google Scholar 

  30. 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, 675–677 (1988)

    Article  Google Scholar 

  31. Malomed, B.A., Mostofi, A., Chu, P.L.: Transformation of a dark soliton into a bright pulse. J. Opt. Soc. Am. B 17, 507–513 (2000)

    Article  Google Scholar 

  32. 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 

  33. Huang, Q.M.: Integrability and dark soliton solutions for a high-order variable coefficients nonlinear Schrödinger equation. Appl. Math. Lett. 93, 29–33 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  34. Wang, L.L., Liu, W.J.: Stable soliton propagation in a coupled \((2+1)\) dimensional Ginzburg–Landau system. Chin. Phys. B 29(7), 070502 (2020)

    Article  Google Scholar 

  35. Cao, Q.H., Dai, C.Q.: Symmetric and anti-symmetric solitons of the fractional second-and third-order nonlinear Schrödinger equation. Chin. Phys. Lett. 38(9), 090501 (2021)

    Article  Google Scholar 

  36. Yan, Y.Y., Liu, W.J.: Soliton rectangular pulses and bound states in a dissipative system modeled by the variable-coefficients complex cubic-quintic Ginzburg–Landau equation. Chin. Phys. Lett. 38(9), 094201 (2021)

    Article  Google Scholar 

  37. Chen, H.H., Lee, Y.C.: Internal-wave solitons of fluids with finite depth. Phys. Rev. Lett. 43, 264–266 (1979)

    Article  MathSciNet  Google Scholar 

  38. Lan, Z.Z., Gao, B.: Lax pair, infinitely many conservation laws and solitons for a \((2+1)\)-dimensional Heisenberg ferromagnetic spin chain equation with time-dependent coefficients. Appl. Math. Lett. 79, 6–12 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  39. Lan, Z.Z.: Conservation laws, modulation instability and solitons interactions for a nonlinear Schrödinger equation with the sextic operators in an optical fiber. Opt. Quant. Electron. 50, 340 (2018)

    Article  Google Scholar 

  40. Lan, Z.Z., Gao, B., Du, M.J.: Dark solitons behaviors for a \((2+1)\)-dimensional coupled nonlinear Schrödinger system in an optical fiber. Chaos, Solitons Fractals 111, 169–174 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  41. Xie, X.Y., Meng, G.Q.: Collisions between the dark solitons for a nonlinear system in the geophysical fluid. Chaos, Solitons Fractals 107, 143–145 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  42. Zhang, Y.J., Zhao, D., Luo, H.G.: The role of middle latency evoked potentials in early prediction of favorable outcomes among patients with severe ischemic brain injuries. Ann. Phys. 350, 112 (2014)

    Article  Google Scholar 

  43. Wu, X.Y., Tian, B., Yin, H.M., Du, Z.: Rogue-wave solutions for a discrete Ablowitz-Ladik equation with variable coefficients for an electrical lattice. Nonlinear Dyn. 93, 1635 (2018)

    Article  MATH  Google Scholar 

  44. Chai, J., Tian, B., Wang, Y.F.: Mixed-type vector solitons for the coupled cubic-quintic nonlinear Schrödinger equations with variable coefficients in an optical fiber. Phys. A 434, 296 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  45. Zhang, Y.J., Yang, C.Y., Yu, W.T., Liu, M.L., Ma, G.L., Liu, W.J.: Some types of dark soliton interactions in inhomogeneous optical fibers. Opt. Quant. Electron. 50, 295–302 (2018)

    Article  Google Scholar 

Download references

Acknowledgements

We acknowledge the financial support from the National Natural Science Foundation of China (Grant Nos. 11875009, 11905009, 12075034).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Qin Zhou.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest concerning the publication of this manuscript.

Ethical approval

The authors declare that they have adhered to the ethical standards of research execution.

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, G., Zhao, J., Zhou, Q. et al. Soliton interaction control through dispersion and nonlinear effects for the fifth-order nonlinear Schrödinger equation. Nonlinear Dyn 106, 2479–2484 (2021). https://doi.org/10.1007/s11071-021-06915-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-021-06915-0

Keywords

Navigation