Skip to main content
SpringerLink
Log in

Stability of soliton solutions for a \(\mathcal {PT}\)- symmetric NLDC considering high-order dispersion and nonlinear effects simultaneously

  • Published:
Optical and Quantum Electronics Aims and scope Submit manuscript

Abstract

In this paper, we analytically solve the coupled equations of a \(\mathcal {PT}\)-Symmetric NLDC by considering high-order dispersion and nonlinear effects (Raman Scattering and self-steeping) simultaneously in normal dispersion regime. To the best of knowledge no works has been done in previous studies to decoupled these equations and obtain an exact analytical solution. The new exact bright solitary solutions are derived. In addition, to study the stability and instability of these propagated solitons in a \(\mathcal {PT}\)-Symmetric NLDC, perturbation theory is used. Numerical methods are applied to find perturbed eigenvalues and eigenfunctions. The Stability of obtained four perturbed eigenvalues and perturbed eigenfunctions for a \(\mathcal {PT}\)-Symmetric NLDC equations regard to high-order effects are examined. Using these results and simulating the propagation of perturbed temporal bright solitons through \(\mathcal {PT}\)-Symmetric NLDC show that perturbed solitons are mostly stable. This means that high-order dispersion and nonlinear effects canceled each other and do not affected the propagated solitons. Furthermore, the evolution of perturbed solitons energies match well the previous results and confirmed the stability of these solitons in a \(\mathcal {PT}\)-Symmetric NLDC. As seen the energies of pulses in bar and cross behave in two manner 1) the exchange of energy is happened in some periods, but the shape of each pulse in bar and cross is preserved. Therefore, the solitons under this eigenfunction perturbation are mostly stable. 2) the evolution of energy in the bar and cross, demonstrate that there is no changes in their energies and they remain constant. It is straightforward to show that in spite of considering high-order effects, the perturbed soliton conserve the shape and it remain stable. The deliverables of this article not only demonstrate a novel approach to ultrafast pulses, solitons and optical couplers, but more fundamentally, they could give insight for improving the new medical equipments technologies, enabling innovations in nonlinear optics and their usage in designing new communication systems and Photonic devices.

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

Similar content being viewed by others

References

  • Ahmad, H., Seadawy, A.R., Khana, T.A.: modified variational iteration algorithm to find approximate solutions of nonlinear Parabolic equation. Mathemat. Comput. Simul. 177, 13–23 (2020)

    Article  Google Scholar 

  • Alexeeva, N.V., Barashenkov, I.V., Sukhorukov, A.A., Kivshar, Y.S.: Optical solitons in PT-Symmetric nonlinear couplers with gain and loss. Phys. Rev. A. 85, 063837 (2012)

    Article  ADS  Google Scholar 

  • Arshad, M., Seadawy, A.R., Lu, D.: Modulation stability and optical soliton solutions of nonlinear Schrödinger equation with higher order dispersion and nonlinear terms and its applications. Superlattices Microstruct. 112, 422–434 (2017). https://doi.org/10.1016/j.spmi.2017.09.054

    Article  ADS  Google Scholar 

  • Bender, C.M., Boettcher, S.: Real spectra in non- Hermitian Hamiltonians having PT- symmetry. Phys. Rev. Lett. 80, 5243–5246 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  • Bender, C.M., Brody, D.C., Jones, H.F.: Complex extension of quantum mechanics. Phys. Rev. Lett. 89, 230401–240404 (2002)

    Article  MathSciNet  Google Scholar 

  • Cheemaa, N., Seadawy, A.R., Chen, S.: More general families of exact solitary wave solutions of the nonlinear Schrödinger equation with their applications in nonlinear optics. Eur. Phys. J. Plus 133, 547 (2018). https://doi.org/10.1140/epjp/i2018-12354-9

    Article  Google Scholar 

  • Cheemaa, N., Seadawy, A.R., Chen, S.: Some new families of solitary wave solutions of the generalized Schamel equation and their applications in plasma physics. Eur. Phys. J. Plus 134, 117 (2019). https://doi.org/10.1140/epjp/i2019-12467-7

    Article  Google Scholar 

  • Chen, X., Yang, J.: A direct perturbation theory for solitons of the derivative nonlinear Schrodinger equation and the modified nonlinear Schrodinger equation. Phys. Rev. E. 65, 066608 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  • Dai, C.Q., Zhang, J.F.: Controllable dynamical behaviors for spatiotemporal bright solitons on continues wave background. Nonlinear Dyn. 2049–2057 (2013)

  • Farah, N., Seadawy, A.R., Ahmad, S., et al.: Interaction properties of soliton molecules and Painleve analysis for nano bioelectronics transmission model. Opt. Quant. Electron 52, 329 (2020). https://doi.org/10.1007/s11082-020-02443-0

    Article  Google Scholar 

  • Govindarajia, A., Mahalingamb, A., Uthayakumar, A.: Femtosecond pulse switching in a fiber coupler with third order dispersion and self-steepening effects. Optik Int. J. Light Electron Opt. 125, 4135–4139 (2014). https://doi.org/10.1016/j.ijleo.2014.01.098

    Article  Google Scholar 

  • Iqbal, M., Seadawy, A.R., Khalil, O.H., Lu, D.: New solitary wave solutions of nonlinear Nizhnik-Novikov-Vesselov equation. Results Phys. 16, 102838 (2020). https://doi.org/10.1016/j.rinp.2019.102838

    Article  Google Scholar 

  • Karlsson, M.: Nonlinear propagation of optical pulses and beams. Chalmers University of Technology, Submitted to the School of Electrical Engineering (1994)

  • Kivshar, Y.S., Agrawal, G.P.: Optical Solitons from fibers to photonic crystals. Academic Press, Sandiego (2003)

    Google Scholar 

  • Liu, W., Pang, L., Han, H., Bi, K., Lei, M., Wei, Z.: Tungsten disulphide for ultrashort pulse generation in all-fiber lasers. Nanoscale 9, 5806–5811 (2017). https://doi.org/10.1039/C7NR00971B

    Article  Google Scholar 

  • Liu, W., Pang, L., Han, H., Liu, M., Lei, M., Fang, S., Teng, H., Wei, Z.: Tungsten disulfide saturable absorbers for 67 fs mode-locked erbium-doped fiber lasers. Opt. Express 25, 2950–2959 (2017). https://doi.org/10.1364/OE.25.002950

    Article  ADS  Google Scholar 

  • Liu, W., Pang, L., Han, H., Shen, Z., Lei, M., Teng, H., Wei, Z.: Dark solitons in WS2 erbium-doped fiber lasers. Photon. Res. 4, 111–114 (2016). https://doi.org/10.1364/PRJ.4.000111

    Article  Google Scholar 

  • Liu, X., Triki, H., Zhou, Q., et al.: Analytic study on interactions between periodic solitons with controllable parameters. Nonlinear Dyn. 94, 703–709 (2018). https://doi.org/10.1007/s11071-018-4387-7

    Article  Google Scholar 

  • Liu, W., Zhu, Y., Liu, M., Wen, B., Fang, S., Teng, H., Lei, M., Liu, L., Wei, Z.: Optical properties and applications for MoS2-Sb2Te3-MoS2 heterostructure materials. Photon. Res. 6, 220–227 (2018)

    Article  Google Scholar 

  • Lu, X., Ma, W.X.: The Inverse Cascade and Nonlinear Alpha-Effect in Simulations of Isotropic Helical Hydromagnetic Turbulence. Nonlinear Dyn. 2755–2758 (2016). https://doi.org/10.1007/s11071

  • Lupu, A., Benisty, H., Degiron, A.: Using optical PT-symmetry for switching applications. Photonics Nanostruct. Fundam. Appl. 12, 305–311 (2014)

    Article  ADS  Google Scholar 

  • Nauman, R., Saima, A., Ahmad, J.: Optical solitons and stability analysis for the generalized second-order nonlinear Schrödinger equation in an optical fiber. Int. J. Nonlinear Sci. Numer. Simul. 21(7–8), 855–863 (2020). https://doi.org/10.1515/ijnsns-2019-0287

    Article  MathSciNet  Google Scholar 

  • Runge, A.F.J., Hudson, D.D., Tam, K.K., Martijin de Stereke, C., Blanco-Redono, A.: The pure-quartic soliton laser. Nat. Photonics 14, 492–497 (2020). https://doi.org/10.1038/s41566-020-0629-6

    Article  Google Scholar 

  • Ruter, C.E., Mrkis, K.G., El-Gnainy, R., Christodoulides, D.N., Segev, M., Kip, D.: Observation of parity-time symmetry in optics. Nat. Phys. 6, 192 (2010)

    Article  Google Scholar 

  • Safaei, L., Hatami, M., Borhani Zarandi, M.: Stability of temporal dark soliton in PT-symmetric nonlinear fiber couplers in normal dispersion regime. J. Optoelectron. Nanostruct. 3, 141 (2016)

    Google Scholar 

  • Safaei, L., Hatami, M., Borhani Zarandi, M.: Numerical analysis of stability for temporal Bright solitons in a PT-symmetric NLDC. J. Optoelectron. Nanostruct. 2(2), 69–78 (2017)

    Google Scholar 

  • Safaei, L., Hatami, M., Borhani Zarandi, M.: PT-symmetric nonlinear directional fiber couplers with gain and loss for ultrashort optical pulses. J. Laser Opt. Photonics. 4, 155 (2017). https://doi.org/10.4172/2469-410X.1000155

    Article  Google Scholar 

  • Safaei, L., Hatami, M., Borhani Zarandi, M.: The effect of relative phase on the stability of temporal dark soliton in PT-Symmetric nonlinear directional fiber coupler. Opt. Quant. Electron. 50, 382 (2018). https://doi.org/10.1007/s11082-018-1646-2

    Article  Google Scholar 

  • Seadawy, A.R.: Two-dimensional interaction of a shear flow with a free surface in a stratified fluid and its solitary-wave solutions via mathematical methods. Euro. Phys. J. Plus 132, 518 (2017). https://doi.org/10.1140/epjp/i2017-11755-6

    Article  Google Scholar 

  • Seadawy, A.: R: Ion acoustic solitary wave solutions of two-dimensional nonlinear Kadomtsev-Petviashvili-Burgers equation in quantum plasma. Mathemat. Methods Appl. Sci. 40(5), 1598–1607 (2017). https://doi.org/10.1002/mma.4081

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Seadawy, A.R., Iqbal, M., Lu, D.: nonlinear wave solutions of the Kudryashov-Sinelshchikov dynamical equation. J. Taibah Univ. Sci. 13, 10601072 (2019). https://doi.org/10.1080/16583655.2019.1680170

    Article  Google Scholar 

  • Suchkov, S.V., Sukhorukov, A.A., Huang, J., Dmitriev, S.V., Lee, C., Kivshar, Y.S.: Nonlinear switching and solitons in PT-Symmetric photonic systems. Laser Photonics Rev. 14 (2015)

  • Sukhorukov, A.A., Xu, Z.Y., Kivshar, Y.S.: Nonlinear suppression of time reversals in PT-symmetric optical couplers. Phys. Rev. A. 82, 043815–043818 (2010)

    Article  ADS  Google Scholar 

  • Wang, C., Nie, Z., Xie, W., Gao, J., Zhou, Q., Liu, W.: Dark soliton control based on dispersion and nonlinearity for third-order nonlinear Schrodinger equation. Optik 184, 370–376 (2019). https://doi.org/10.1016/j.ijleo.2019.04.020

    Article  ADS  Google Scholar 

  • Wang, Y.Y., Zhang, Y.P., Dai, C.Q.: Re-study on localized structures based on variable separation solutions from the modified tanh-function method. Nonlinear Dyn. 83, 1331 (2016). https://doi.org/10.1007/s11071-015-2406-5

    Article  MathSciNet  Google Scholar 

  • Zhu, Y., Qin, W., Li, J., et al.: Recurrence behavior for controllable excitation of rogue waves in a two-dimensional PT-Symmetric couple. Nonlinear Dyn. 88, 1883 (2017)

    Article  Google Scholar 

  • Zyablovsky, A.A., Vinogradov, A.P., Pukhov, A.A., Dorofeenko, A.V., Lisyansky, A.A.: PT-symmetry in optics. Phys. Usp. 57, 1063–1082 (2014)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Science, Department of Physics, Yazd University, Yazd, Iran

    Lida Safaei & Mahmood Borhani Zarndi

  2. Faculty of Physics, Shiraz University of Technology, Shiraz, Iran

    Mohsen Hatami

Authors
  1. Lida Safaei
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mohsen Hatami
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mahmood Borhani Zarndi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lida Safaei.

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

Safaei, L., Hatami, M. & Zarndi, M.B. Stability of soliton solutions for a \(\mathcal {PT}\)- symmetric NLDC considering high-order dispersion and nonlinear effects simultaneously. Opt Quant Electron 53, 414 (2021). https://doi.org/10.1007/s11082-021-03059-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11082-021-03059-8

Keywords

Search

Navigation