Skip to main content
SpringerLink
Log in

Phase shift, oscillation and collision of the anti-dark solitons for the (3+1)-dimensional coupled nonlinear Schrödinger equation in an optical fiber communication system

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

Abstract

In the long-distance optical fiber communication system, we consider the (3+1)-dimensional coupled nonlinear Schrödinger equation with perturbation functions, which controls the transverse effects, because of the complexity of nonlinear phenomena. Anti-dark two-soliton solutions are derived by the Hirota method. The perturbation function representing the dissipation rate is discussed. Five different transmission structures of solitons are presented. Besides, their propagation and collision dynamics are analyzed. Control methods for phase shift and oscillation are also suggested. We hope results in this paper are of guiding significance to the research of phase shifters, optical logic devices and ultrashort pulse lasers.

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

  1. 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 

  2. 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 

  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. 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 

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

    Article  MathSciNet  Google Scholar 

  6. Tao, M.S., Zhang, N., Gao, D.Z., Yang, H.W.: Symmetry analysis for three-dimensional dissipation Rossby waves. Adv. Differ. Equ. 2018, 300 (2018)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. Gu, J.Y., Zhang, Y., Dong, H.H.: Dynamic behaviors of interaction solutions of (3+1)-dimensional shallow mater wave equation. Comput. Math. Appl. 76(6), 1408–1419 (2018)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  10. Liu, Y., Dong, H.H., Zhang, Y.: Solutions of a discrete integrable hierarchy by straightening out of its continuous and discrete constrained flows. Anal. Math. Phys. 2018, 1–17 (2018)

    Google Scholar 

  11. Wazwaz, A.M.: A study on a two-wave mode Kadomtsev–Petviashvili equation: conditions for multiple soliton solutions to exist. Math. Method. Appl. Sci. 40, 4128–4133 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  12. Guo, M., Zhang, Y., Wang, M., Chen, Y.D., Yang, H.W.: A new ZK-ILW equation for algebraic gravity solitary waves in finite depth stratified atmosphere and the research of squall lines formation mechanism. Comput. Math. Appl. 143, 3589–3603 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  13. Yang, H.W., Chen, X., Guo, M., Chen, Y.D.: A new ZK-BO equation for three-dimensional algebraic Rossby solitary waves and its solution as well as fission property. Nonlinear Dyn. 91, 2019–2032 (2018)

    Article  MATH  Google Scholar 

  14. Zhao, B.J., Wang, R.Y., Sun, W.J., Yang, H.W.: Combined ZK-mZK equation for Rossby solitary waves with complete Coriolis force and its conservation laws as well as exact solutions. Adv. Differ. Equ. 2018, 42 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  15. Lu, C., Fu, C., Yang, H.W.: Time-fractional generalized Boussinesq equation for Rossby solitary waves with dissipation effect in stratified fluid and conservation laws as well as exact solutions. Appl. Math. Comput. 327, 104–116 (2018)

    MathSciNet  MATH  Google Scholar 

  16. Zhang, C., Liu, J., Fan, X.W., Peng, Q.Q., Guo, X.S., Jiang, D.P., Qian, X.B., Su, L.B.: Compact passive Q-switching of a diode-pumped Tm, Y: \(\text{ CaF }_{2}\) laser near 2 \(\iota \)m. Opt. Laser Technol. 103, 89–92 (2018)

    Article  Google Scholar 

  17. Liu, X.Y., Triki, H., Zhou, Q., Liu, W.J., Biswas, A.: Analytic study on interactions between periodic solitons with controllable parameters. Nonlinear Dyn. 94, 1703–709 (2018)

    Article  Google Scholar 

  18. Liu, J., Wang, Y.G., Qu, Z.S., Fan, X.W.: 2 \(\mu \)m passive Q-switched mode-locked \(\text{ Tm }^{3+}\): \(YAP\) laser with single-walled carbon nanotube absorber. Opt. Laser Technol. 44(4), 960–962 (2012)

    Article  Google Scholar 

  19. Zhang, Y.J., Yang, C.Y., Yu, W.T., Mirzazadeh, M., Zhou, Q., Liu, W.J.: Interactions of vector anti-dark solitons for the coupled nonlinear Schrödinger equation in inhomogeneous fibers. Nonlinear Dyn. 94, 1351–1360 (2018)

    Article  Google Scholar 

  20. Lin, M.X., Peng, Q.Q., Hou, W., Fan, X.W., Liu, J.: 1.3 \(\iota \)m Q-switched solid-state laser based on few-layer \(\text{ ReS }_{2}\) saturable absorber. Opt. Laser Technol. 109, 90–93 (2019)

    Article  Google Scholar 

  21. Ali, K., Rizvi, S.T.R., Nawaz, B., Younis, M.: Optical solitons for paraxial wave equation in Kerr media. Mod. Phys. Lett. B 33(3), 1950020 (2019)

    Article  MathSciNet  Google Scholar 

  22. Younis, B., Younis, M., Ahmed, M.O., Rizvi, S.T.R.: Chriped optical solitons in nanofibers. Mod. Phys. Lett. B 32(26), 1850320 (2018)

    Article  Google Scholar 

  23. Ali, K., Rizvi, S.T.R., Khalil, A., Younis, M.: Chriped and dipole soliton in nonlinear negative-index materials. Optik 172, 657–661 (2018)

    Article  Google Scholar 

  24. Liu, W.J., Zhang, Y.J., Triki, H., Mirzazadeh, M., Ekici, M., Zhou, Q., Biswas, A., Belic, M.: Interaction properties of solitonics in inhomogeneous optical fibers. Nonlinear Dyn. 95, 557–563 (2019)

    Article  Google Scholar 

  25. Zhang, F., Wu, Y.J., Liu, J., Pang, S.Y., Ma, F.K., Jiang, D.P., Wu, Q.H., Su, L.B.: Mode locked \(\text{ Nd }^{3+}\) and \(\text{ Gd }^{3+}\) co-doped calcium fluoride crystal laser at dual gain lines. Opt. Laser Technol. 100, 294–297 (2018)

    Article  Google Scholar 

  26. Liu, X.Y., Triki, H., Zhou, Q., Mirzazadeh, M., Liu, W.J., Biswas, A., Belic, M.: Generation and control of multiple solitons under the influence of parameters. Nonlinear Dyn. 95, 143–150 (2019)

    Article  Google Scholar 

  27. Zhang, F., Liu, J., Li, W.W., Mei, B.C., Jiang, D.P., Qian, X.B., Su, L.B.: Dual-wavelength continuous-wave and passively Q-switched Nd, Y: \(\text{ SrF }_{2}\) ceramic laser. Opt. Eng. 55(10), 106114 (2016)

    Article  Google Scholar 

  28. Yang, C.Y., Liu, W.J., Zhou, Q., Mihalache, D., Malomed, B.A.: One-soliton shaping and two-soliton interaction in the fifth-order variable-coefficient nonlinear Schrödinger equation. Nonlinear Dyn. 95, 369–380 (2019)

    Article  Google Scholar 

  29. Wu, Y.J., Zhang, C., Liu, J.J., Zhang, H.N., Yang, J.M., Liu, J.: Silver nanorods absorbers for Q-switched Nd: YAG ceramic laser. Opt. Laser Technol. 97, 268–271 (2017)

    Article  Google Scholar 

  30. Liu, W.J., Liu, M.L., Liu, B., Quhe, R.G., Lei, M., Fang, S.B., Teng, H., Wei, Z.Y.: Nonlinear optical properties of \(\text{ MoS }_{2}-\text{ WS }_{2}\) heterostructure in fiber lasers. Opt. Express 27, 6689–6699 (2019)

    Article  Google Scholar 

  31. Li, C., Fan, M.W., Liu, J., Su, L.B., Jiang, D.P., Qian, X.B., Xu, J.: Operation of continuous wave and Q-switching on diode-pumped Nd, Y: \(\text{ CaF }_{2}\) disordered crystal. Opt. Laser Technol. 69, 140–143 (2015)

    Article  Google Scholar 

  32. Li, L., Lv, R.D., Liu, S.C., Chen, Z.D., Wang, J., Wang, Y.G., Ren, W., Liu, W.J.: Ferroferric-oxide nanoparticle based Q-switcher for a 1 \(\iota \)m region. Opt. Mater. Express 9, 731–738 (2019)

    Article  Google Scholar 

  33. Cai, W., Peng, Q.Q., Hou, W., Liu, J., Wang, Y.G.: Picosecond passively mode-locked laser of 532 nm by reflective carbon nanotube. Opt. Laser Technol. 58, 194–196 (2014)

    Article  Google Scholar 

  34. Li, L., Lv, R.D., Wang, J., Chen, Z.D., Wang, H.Z., Liu, S.C., Ren, W., Liu, W.J., Wang, Y.G.: Optical nonlinearity of \(\text{ ZrS }_{2}\) and applications in fiber laser. Nanomaterials 9, 315 (2019)

    Article  Google Scholar 

  35. Wang, Y.G., Qu, Z.S., Liu, J., Tsang, Y.H.: Graphene oxide absorbers for watt-level high-power passive mode-locked Nd: \(\text{ GdVO }_{4}\) laser operating at 1 \(\iota \)m. J. Lightwave Technol. 30(20), 3259–3262 (2012)

    Article  Google Scholar 

  36. Lei, M.Z., Zheng, Z.N., Qian, J.W., Xie, M.T., Bai, Y.P., Gao, X.L., Huang, S.G.: Broadband chromatic-dispersion-induced power-fading compensation for radio-over-fiber links based on Hilbert transform. Opt. Lett. 44, 155–158 (2019)

    Article  Google Scholar 

  37. Zhu, H.T., Zhao, L.N., Liu, J., Xu, S.C., Cai, W., Jiang, S.Z., Zheng, L.H., Su, L.B., Xu, J.: Monolayer graphene saturable absorber with sandwich structure for ultrafast solid-state laser. Opt. Eng. 55(8), 081304 (2016)

    Article  Google Scholar 

  38. Xie, M.T., Zhao, M.Y., Lei, M.Z., Wu, Y.L., Liu, Y.A., Gao, X.L., Huang, S.G.: Anti-dispersion phase-tunable microwave mixer based on a dual-drive dual-parallel Mach–Zehnder modulator. Opt. Express 26, 454–462 (2018)

    Article  Google Scholar 

  39. Cai, W., Jiang, S.Z., Xu, S.C., Li, Y.Q., Liu, J., Li, C., Zheng, L.H., Su, L.B., Xu, J.: Graphene saturable absorber for diode pumped Yb: \(\text{ Sc }_{2}\text{ SiO }_{5}\) mode-locked laser. Opt. Laser Technol. 65, 1–4 (2015)

    Article  Google Scholar 

  40. Gao, X.L., Zhao, M.Y., Xie, M.T., Lei, M.Z., Song, X.Y., Bi, K., Zheng, Z.N., Huang, S.G.: 2D optical-controlled radio frequency orbital angular momentum beam steering system based on dual-parallel Mach-Zehnder modulator. Opt. Lett. 44, 255–258 (2019)

    Article  Google Scholar 

  41. Zhu, H.T., Liu, J., Jiang, S.Z., Xu, S.C., Su, L.B., Jiang, D.P., Qian, X.B., Xu, J.: Diode-pumped Yb, Y: \(\text{ CaF }_{2}\) laser mode-locked by monolayer graphene. Opt. Laser Technol. 75, 83–86 (2015)

    Article  Google Scholar 

  42. Guo, B.L., Huang, S.G., Shang, Y., Zhang, Y.Q., Li, W.Z., Yin, S., Zhang, Y.J.: Timeslot switching-based optical bypass in data center for intra-rack elephant flow with an ultrafast DPDK-enabled timeslot allocator. J. Lightwave Technol. 37, 2253–2260 (2019)

    Article  Google Scholar 

  43. Li, X., Zhang, L., Tang, Y., Gao, T., Zhang, Y.J., Huang, S.G.: On-demand routing, modulation level and spectrum allocation (OD-RMSA) for multicast service aggregation in elastic optical networks. Opt. Express 26, 24506–24530 (2018)

    Article  Google Scholar 

  44. Medina, A.C., Schmid, R.: Solution of high order compact discretized 3D elliptic partial differential equations by an accelerated multigrid method. J. Comput. Appl. Math. 350, 343–352 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  45. Sagis, L.M.C.: Coarse-grained models for diffusion in oil-filled hydrogel microbeads. Food Hydrocoll. 89, 294–301 (2019)

    Article  Google Scholar 

  46. Tambue, A., Mukam, J.D.: Strong convergence of the linear implicit Euler method for the finite element discretization of semilinear SPDEs driven by multiplicative or additive noise. Appl. Math. Comput. 346, 23–40 (2019)

    MathSciNet  Google Scholar 

  47. Wang, Y.Y., Dai, C.Q., Xu, Y.Q., Zheng, J., Fan, Y.: Dynamics of nonlocal and localized spatiotemporal solitons for a partially nonlocal nonlinear Schrödinger equation. Nonlinear Dyn. 92(3), 1261–1269 (2018)

    Article  Google Scholar 

  48. He, C., Tang, Y., Ma, J.: New interaction solutions for the (3+1)-dimensional Jimbo-Miwa equation. Comput. Math. Appl. 76(9), 2141–2147 (2018)

    Article  MathSciNet  Google Scholar 

  49. Liu, J.G., You, M.X., Zhou, L., Ai, G.P.: The solitary wave, rogue wave and periodic solutions for the (3+1)-dimensional soliton equation. Z. Angew. Math. Phys. 70(1), 4 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  50. Pandey, M.: Unparticle decay of neutrinos and its possible signatures at \(Km^{2}\) detector for (3+1) flavour framework. J. High Energy Phys. 2019(1), 66 (2019)

    Article  Google Scholar 

  51. Rashed, A.S.: Analysis of (3+1)-dimensional unsteady gas flow using optimal system of Lie symmetries. Math. Comput. Simul. 156, 327–346 (2019)

    Article  MathSciNet  Google Scholar 

  52. Shokri, M., Sadooghi, N.: Evolution of magnetic fields from the 3+1 dimensional self-similar and Gubser flows in ideal relativistic magnetohydrodynamics. J. High Energy Phys. 2018(11), 181 (2018)

    Article  MathSciNet  Google Scholar 

  53. Yuan, Y.Q., Tian, B., Liu, L., Chai, H.P., Sun, Y.: Semi-rational solutions for the (3+1)-dimensional Kadomtsev–Petviashvili equation in a plasma or fluid. Comput. Math. Appl. 76(11–12), 2566–2574 (2018)

    Article  MathSciNet  Google Scholar 

  54. Yue, Y., Huang, L., Chen, Y.: Localized waves and interaction solutions to an extended (3+1)-dimensional Jimbo–Miwa equation. Appl. Math. Lett. 89, 70–77 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  55. Chen, S., Zheng, X., Zhan, Y., Ma, S.D., Deng, D.M.: Propagation properties of chirped Airy hollow Gaussian wave packets. Opt. Commun. 435, 164–172 (2019)

    Article  Google Scholar 

  56. Zhu, H.P., Qian, L.R.: Bright and dark wirelike spatiotemporal solitons of a partially nonlocal nonlinear Schrödinger equation. Appl. Math. Lett. 82, 118–125 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  57. Yong, X., Li, X., Huang, Y.: General lump-type solutions of the (3+1)-dimensional Jimbo–Miwa equation. Appl. Math. Lett. 86, 222–228 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  58. Liu, X.Z., Yu, J., Lou, Z.M., Qian, X.M.: Residual symmetry analysis and CRE integrability of the (3+1)-dimensional Burgers system. Eur. Phys. J. Plus 133(12), 503 (2018)

    Article  Google Scholar 

  59. Ma, W.X.: Abundant lumps and their interaction solutions of (3+1)-dimensional linear PDEs. J. Geom. Phys. 133, 10–16 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  60. Sun, B., Wazwaz, A.M.: General high-order breathers and rogue waves in the (3+1)-dimensional KP-Boussinesq equation. Commun. Nonlinear Sci. Numer. Simul. 64, 1–13 (2018)

    Article  MathSciNet  Google Scholar 

  61. Tariq, K.U., Seadawy, A.R., Alamri, S.Z.: Computational soliton solutions to (3+1)-dimensional generalised Kadomtsev–Petviashvili and (2+1)-dimensional Gardner-Kadomtsev-Petviashvili models and their applications. Pramana-J. Phys. 91(5), 68 (2018)

    Article  Google Scholar 

  62. Verma, P., Kaur, L.: Integrability, bilinearization and analytic study of new form of (3+1)-dimensional B-type Kadomstev–Petviashvili (BKP)-Boussinesq equation. Appl. Math. Comput. 346, 879–886 (2019)

    MathSciNet  Google Scholar 

  63. Wang, Y.Y., Dai, C.Q., Xu, Y.Q., Zheng, J., Fan, Y.: Dynamics of nonlocal and localized spatiotemporal solitons for a partially nonlocal nonlinear Schrödinger equation. Nonlinear Dyn. 92(3), 1261–1269 (2018)

    Article  Google Scholar 

  64. Mansfield, E.L., Reid, G.J., Clarkson, P.A.: Nonclassical reductions of a 3+1-cubic nonlinear Schrödinger system. Comput. Phys. Comun. 115(2–3), 460–488 (1998)

    Article  MATH  Google Scholar 

  65. Huang, Z.R., Tian, B., Wang, Y.P., Sun, Y.: Bright soliton solutions and collisions for a (3+1)-dimensional coupled nonlinear Schrödinger system in optical-fiber communication. Comput. Math. Appl. 69(12), 1383–1389 (2015)

    Article  MathSciNet  Google Scholar 

  66. Hirota, R.: Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons. Phys. Rev. Lett. 27, 1192–1194 (1971)

    Article  MATH  Google Scholar 

Download references

Acknowledgements

We acknowledge the financial support from the National Natural Science Foundation of China (Grant Nos. 11674036 and 11875008), by the Beijing Youth Top-notch Talent Support Program (Grant No. 2017000026833ZK08), and by the Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications, Grant No. IPOC2017ZZ05).

Author information

Authors and Affiliations

  1. State Key Laboratory of Information Photonics and Optical Communications, and School of Science, Beijing University of Posts and Telecommunications, P. O. Box 122, Beijing, 100876, People’s Republic of China

    Weitian Yu & Wenjun Liu

  2. Radiation Physics Laboratory, Department of Physics, Faculty of Sciences, Badji Mokhtar University, P. O. Box 12, 23000, Annaba, Algeria

    Houria Triki

  3. School of Electronics and Information Engineering, Wuhan Donghu University, Wuhan, 430212, People’s Republic of China

    Qin Zhou

  4. Department of Physics, Chemistry and Mathematics, Alabama A & M University, Normal, AL, 35762, USA

    Anjan Biswas

  5. Department of Mathematics, King Abdulaziz University, Jeddah, 21589, Saudi Arabia

    Anjan Biswas

  6. Department of Mathematics and Statistics, Tshwane University of Technology, Pretoria, 0008, South Africa

    Anjan Biswas

Authors
  1. Weitian Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wenjun Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Houria Triki
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Qin Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Anjan Biswas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wenjun Liu.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

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

Yu, W., Liu, W., Triki, H. et al. Phase shift, oscillation and collision of the anti-dark solitons for the (3+1)-dimensional coupled nonlinear Schrödinger equation in an optical fiber communication system. Nonlinear Dyn 97, 1253–1262 (2019). https://doi.org/10.1007/s11071-019-05045-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-019-05045-y

Keywords

Search

Navigation