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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

  • Weitian Yu
  • Wenjun LiuEmail author
  • Houria Triki
  • Qin Zhou
  • Anjan Biswas
Original Paper
  • 16 Downloads

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.

Keywords

Anti-dark solitons Soliton collision Analytic solution Hirota method Nonlinear Schrödinger equation 

Notes

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).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. 1.
    Wazwaz, A.M.: A two-mode modified KdV equation with multiple soliton solutions. Appl. Math. Lett. 70, 1–6 (2017)MathSciNetzbMATHGoogle Scholar
  2. 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)MathSciNetzbMATHGoogle Scholar
  3. 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)MathSciNetGoogle Scholar
  4. 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)MathSciNetzbMATHGoogle Scholar
  5. 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)MathSciNetGoogle Scholar
  6. 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)MathSciNetzbMATHGoogle Scholar
  7. 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)MathSciNetzbMATHGoogle Scholar
  8. 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)MathSciNetGoogle Scholar
  9. 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)MathSciNetzbMATHGoogle Scholar
  10. 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. 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)MathSciNetzbMATHGoogle Scholar
  12. 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)MathSciNetGoogle Scholar
  13. 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)zbMATHGoogle Scholar
  14. 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)MathSciNetzbMATHGoogle Scholar
  15. 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)MathSciNetGoogle Scholar
  16. 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)Google Scholar
  17. 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)Google Scholar
  18. 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)Google Scholar
  19. 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)Google Scholar
  20. 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)Google Scholar
  21. 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)MathSciNetGoogle Scholar
  22. 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)Google Scholar
  23. 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)Google Scholar
  24. 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)Google Scholar
  25. 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)Google Scholar
  26. 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)Google Scholar
  27. 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)Google Scholar
  28. 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)Google Scholar
  29. 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)Google Scholar
  30. 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)Google Scholar
  31. 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)Google Scholar
  32. 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)Google Scholar
  33. 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)Google Scholar
  34. 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)Google Scholar
  35. 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)Google Scholar
  36. 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)Google Scholar
  37. 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)Google Scholar
  38. 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)Google Scholar
  39. 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)Google Scholar
  40. 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)Google Scholar
  41. 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)Google Scholar
  42. 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)Google Scholar
  43. 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)Google Scholar
  44. 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)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Sagis, L.M.C.: Coarse-grained models for diffusion in oil-filled hydrogel microbeads. Food Hydrocoll. 89, 294–301 (2019)Google Scholar
  46. 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)MathSciNetGoogle Scholar
  47. 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)Google Scholar
  48. 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)MathSciNetGoogle Scholar
  49. 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)MathSciNetzbMATHGoogle Scholar
  50. 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)Google Scholar
  51. 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)MathSciNetGoogle Scholar
  52. 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)MathSciNetGoogle Scholar
  53. 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)MathSciNetGoogle Scholar
  54. 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)MathSciNetzbMATHGoogle Scholar
  55. 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)Google Scholar
  56. 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)MathSciNetzbMATHGoogle Scholar
  57. 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)MathSciNetzbMATHGoogle Scholar
  58. 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)Google Scholar
  59. 59.
    Ma, W.X.: Abundant lumps and their interaction solutions of (3+1)-dimensional linear PDEs. J. Geom. Phys. 133, 10–16 (2018)MathSciNetzbMATHGoogle Scholar
  60. 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)MathSciNetGoogle Scholar
  61. 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)Google Scholar
  62. 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)MathSciNetGoogle Scholar
  63. 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)Google Scholar
  64. 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)zbMATHGoogle Scholar
  65. 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)MathSciNetGoogle Scholar
  66. 66.
    Hirota, R.: Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons. Phys. Rev. Lett. 27, 1192–1194 (1971)zbMATHGoogle Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.State Key Laboratory of Information Photonics and Optical Communications, and School of ScienceBeijing University of Posts and TelecommunicationsBeijingPeople’s Republic of China
  2. 2.Radiation Physics Laboratory, Department of Physics, Faculty of SciencesBadji Mokhtar UniversityAnnabaAlgeria
  3. 3.School of Electronics and Information EngineeringWuhan Donghu UniversityWuhanPeople’s Republic of China
  4. 4.Department of Physics, Chemistry and MathematicsAlabama A & M UniversityNormalUSA
  5. 5.Department of MathematicsKing Abdulaziz UniversityJeddahSaudi Arabia
  6. 6.Department of Mathematics and StatisticsTshwane University of TechnologyPretoriaSouth Africa

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