Abstract
Dark solitons are the subject of intense theoretical and experimental studies in nonlinear optics due to their unique characteristics compared with bright solitons. In this paper, the variable coefficient high-order nonlinear Schrödinger equation in the inhomogeneous optical fiber is investigated. Via the Hirota bilinear method and symbolic computation, the analytic dark two-soliton solutions are obtained. With the suitable choices of functions and coefficients for the obtained dark two-soliton solutions, some new phenomena are presented for the first time. The influences on phases and amplitudes of soliton interactions are detailed analyzed. Moreover, sets of double-triangle structures and methods of changing the propagation direction of dark solitons are introduced. Finally, by choosing suitable functions of the fourth-order dispersion parameter, the arch-structure and M-structure interactions are revealed. Results may be potentially useful in designing all-optical switches and optical fibers.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abdullaev, F.K., Garnier, J.: Dynamical stabilization of solitons in cubic–quintic nonlinear Schrödinger model. Phys. Rev. E 72, 035603R (2005)
Agrawal, G.P.: Nonlinear Fiber Optics. Academic Press, San Diego (2007)
Ankiewicz, A., Akhmediev, N.: Higher-order integrable evolution equation and its soliton solutions. Phys. Lett. A 378, 358–361 (2014)
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)
Chen, C.J., Wai, P.K.A., Menyuk, C.R.: Soliton switch using birefringent optical fibers. Opt. Lett. 15, 477–479 (1990)
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)
Emmanuel, Y., Gholam-Ali, Z.: N-soliton interactions in an extended Schrödinger equation with higher order of nonlinearities. Phys. B 483, 26–36 (2016)
Hasegawa, A., Matsumoto, M.: Optical Solitons in Fibers. Springer, Berlin (2003)
Hasegawa, A., Tappert, F.: Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion. Appl. Phys. Lett. 23, 142–144 (1973)
Hirota, R.: The Direct Method in Soliton Theory, Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (2004)
Kivshar, Y.S., Agrawal, G.P.: Optical Solitons: From Fibers to Photonic Crystals. Academic Press, San Diego (2003)
Kivshar, Y.S., Haelterman, M., Emplit, P., Hamaide, J.P.: Gordon–Haus effect on dark solitons. Opt. Lett. 19, 19–21 (1994)
Lamb Jr., G.L.: Elements of Soliton Theory. Wiley, New York (1980)
Li, W.Y., Ma, G.L., Yu, W.T., Zhang, Y.J., Liu, M.L., Yang, C.Y., Liu, W.J.: Soliton structures in the (\(1+1\))-dimensional Ginzburg–Landau equation with a parity-time-symmetric potential in ultrafast optics. Chin. Phys. B 27, 030504 (2018)
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 \({\rm Sb}_2{\rm Te}_3\) saturable absorbers. Opt. Express 23, 26023–26031 (2015)
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 (2017a)
Liu, W.J., Yu, W.T., Yang, C.Y., Liu, M.L., Zhang, Y.J., Lei, M.: Analytic solutions for the generalized complex Ginzburg–Landau equation in fiber lasers. Nonlinear Dyn. 89, 2933–2939 (2017b)
Liu, W.J., Pang, L.H., Han, H.N., Bi, K., Lei, M., Wei, Z.Y.: Tungsten disulphide for ultrashort pulse generation in all-fiber lasers. Nanoscale 9, 5806–5811 (2017c)
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 (2017d)
Liu, W.J., Liu, M.L., Yin, J.D., Chen, H., Lu, W., Fang, S.B., Teng, H., Lei, M., Yan, P.G., Wei, Z.Y.: Tungsten diselenide for all-fiber lasers with the chemical vapor deposition method. Nanoscale 10, 7971–7977 (2018a)
Liu, W.J., Zhu, Y.N., Liu, M.L., Wen, B., Fang, S.B., Teng, H., Lei, M., Liu, L.M., Wei, Z.Y.: Optical properties and applications for \(\text{MoS}_{2}{-}\text{Sb}_{2}\text{Te}_{3}{-}\text{MoS}_{2}\) heterostructure materials. Photon. Res. 6, 220–227 (2018b)
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 (2018c)
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 (2018d)
Locati, F.S., Romagnoli, M., Tajani, A., Wabnitz, S.: Adiabatic femtosecond soliton active nonlinear directional coupler. Opt. Lett. 17, 1394–1396 (1992)
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)
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)
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)
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)
Wang, Y.F., Tian, B., Sun, W.R.: Analytic study on an extended higher-order nonlinear Schrödinger equation for heisenberg ferromagnet. Chin. J. Phys. 53, 1–11 (2015)
Weiner, A.M., Heritage, J.P., Hawkins, R.J., Thurston, R.N., Kirschner, E.M., Leaird, D.E., Tomlinson, W.J.: Experimental observation of the fundamental dark soliton in optical fibers. Phys. Rev. Lett. 61, 2445–2448 (1988)
Wilson, J., Stegeman, G.I., Wright, E.M.: Soliton switching in an erbium-doped nonlinear fiber coupler. Opt. Lett. 16, 1653–1655 (1991)
Yang, C.Y., Li, W.Y., Yu, W.T., Liu, M.L., Zhang, Y.J., Ma, G.L., Lei, M., Liu, W.J.: Amplification, reshaping, fission and annihilation of optical solitons in dispersion-decreasing fiber. Nonlinear Dyn. 92, 203–213 (2018)
Zhang, H., Tang, D.Y., Zhao, L.M., Knize, R.J.: Vector dark domain wall solitons in a fiber ring laser. Opt. Express 18, 4428–4433 (2010)
Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grant No. 11674036), 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 Nos. IPOC2016ZT04 and IPOC2017ZZ05).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhang, Y., Yang, C., Yu, W. et al. Some types of dark soliton interactions in inhomogeneous optical fibers. Opt Quant Electron 50, 295 (2018). https://doi.org/10.1007/s11082-018-1560-7
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11082-018-1560-7