Nonlinear Dynamics

, Volume 78, Issue 1, pp 755–770 | Cite as

Soliton interactions for coupled nonlinear Schrödinger equations with symbolic computation

  • Wen-Jun Liu
  • Nan Pan
  • Long-Gang Huang
  • Ming Lei
Original Paper


Soliton interactions for the coupled nonlinear Schrödinger equations, governing the propagation of envelopes of electromagnetic waves in birefringent optical fibers, are investigated with symbolic computation. Based on the Hirota method, analytic two- and three-soliton solutions for this model are derived. Relevant interaction properties are discussed. Stationary bound vector solitons with the periodic attraction and repulsion are obtained. Soliton intensity could be reduced if the nonlinearity in optical fibers is enlarged, while the soliton period could be prolonged as the group velocity dispersion in the anomalous dispersion regime of optical fibers increases. Through the asymptotic analysis for the two-soliton solutions, interactions between two solitons are proven to be elastic. Besides, parallel soliton transmission systems without soliton interactions are presented. Moreover, interactions between the regular and bound vector solitons are studied. Dual complex structures and triple-soliton bound states are presented. Results could be of certain value to the studies on the soliton control and optical switching technologies.


Coupled nonlinear Schrödinger equations Symbolic computation Soliton solution Bound vector solitons Soliton interactions 



We express our sincere thanks to the Editors and Referees for their valuable comments. This work has been supported by the National Natural Science Foundation of China (NSFC) (Grant Nos. 61205064), by the Visiting Scholar Funds of the Key Laboratory of Optoelectronic Technology & Systems under Grant No. \(0902011812401_{-}5\), Chongqing University.


  1. 1.
    Agrawal, G.P.: Nonlinear Fiber Optics, 4th edn. Academic, San Diego (2007)Google Scholar
  2. 2.
    Han, S.H., Park, Q.H.: Effect of self-steepening on optical solitons in a continuous wave background. Phys. Rev. E 83, 066601 (2011)CrossRefGoogle Scholar
  3. 3.
    Yang, Z.Y., Zhao, L.C., Zhang, T., Feng, X.Q., Yue, R.H.: Dynamics of a nonautonomous soliton in a generalized nonlinear Schrödinger equation. Phys. Rev. E 83, 066602 (2011)CrossRefGoogle Scholar
  4. 4.
    Biswas, A., Fessak, M., Johnson, S., Beatrice, S., Milovic, D., Jovanoski, Z., Kohl, R., Majid, F.: Optical soliton perturbation in non-Kerr law media: traveling wave solution. Opt. Laser Technol. 44, 263–268 (2012)CrossRefGoogle Scholar
  5. 5.
    Borovkova, O.V., Kartashov, Y.V., Torner, L., Malomed, B.A.: Bright solitons from defocusing nonlinearities. Phys. Rev. E 84, 035602(R) (2011)CrossRefGoogle Scholar
  6. 6.
    Molchan, M.A.: Nonlocal solitons in the parametrically driven nonlinear Schrödinger equation: stability analysis. Phys. Rev. E 84, 056603 (2011)CrossRefGoogle Scholar
  7. 7.
    Petrović, N.Z., Belić, M., Zhong, W.P.: Exact traveling-wave and spatiotemporal soliton solutions to the generalized (3+1)-dimensional Schrödinger equation with polynomial nonlinearity of arbitrary order. Phys. Rev. E 83, 026604 (2011)CrossRefGoogle Scholar
  8. 8.
    Tian, Q., Wu, L., Zhang, J.F., Malomed, B.A., Mihalache, D., Liu, W.M.: Exact soliton solutions and their stability control in the nonlinear Schrödinger equation with spatiotemporally modulated nonlinearity. Phys. Rev. E 83, 016602 (2011)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Guo, R., Hao, H.Q., Zhang, L.L.: Dynamic behaviors of the breather solutions for the AB system in fluid mechanics. Nonlinear Dyn. 74, 701–709 (2013)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Dai, C.Q., Zhu, H.P.: Superposed Kuznetsov-Ma solitons in a two-dimensional graded-index grating waveguide. J. Opt. Soc. Am. B 30, 3291–3297 (2013)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Dai, C.Q., Zhu, H.P.: Superposed Akhmediev breather of the (3 + 1)-dimensional generalized nonlinear Schrödinger equation with external potentials. Ann. Phys. 341, 142–152 (2014)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Guo, R., Hao, H.Q.: Breathers and localized solitons for the Hirota-Maxwell-Bloch system on constant backgrounds in erbium doped fibers. Ann. Phys. 344, 10–16 (2014)CrossRefGoogle Scholar
  13. 13.
    Liu, W.J., Tian, B., Lei, M.: Elastic and inelastic interactions between optical spatial solitons in nonlinear optics. Laser Phys. 23, 095401 (2013)CrossRefGoogle Scholar
  14. 14.
    Zhu, H.P.: Nonlinear tunneling for controllable rogue waves in two dimensional graded-index waveguides. Nonlinear Dyn. 72, 873–882 (2013)CrossRefGoogle Scholar
  15. 15.
    Biswas, A., Konar, S.: Quasi-particle theory of optical soliton interaction. Commun. Nonlinear Sci. Numer. Simul. 12, 1202–1228 (2007)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Biswas, A., Khan, K.R., Rahman, A., Yildirim, A., Hayat, T., Aldossary, O.M.: Bright and dark optical solitons in birefringent fibers with Hamiltonian perturbations and Kerr law nonlinearity. J. Optoelectron. Adv. Mater. 14, 571–576 (2012)Google Scholar
  17. 17.
    Savescu, M., Alshaery, A.A., Bhrawy, A.H., Hilal, E.M., Moraru, L., Biswas, A.: Optical solitons in birefringent fibers with coupled Hirota equation and spatio-temporal dispersion. Wulfenia 21, 35–43 (2014)Google Scholar
  18. 18.
    Savescu, M., Bhrawy, A.H., Alshaery, A.A., Hilal, E.M., Khan, K.R., Mahmood, M.F., Biswas, A.: Optical solitons in nonlinear directional couplers with spatio-temporal dispersion. J. Mod. Opt. 61, 441–458 (2014)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Bhrawy, A.H., Alshaery, A.A., Hilal, E.M., Manrakhan, W.N., Savescu, M., Biswas, A.: Dispersive optical solitons with Schrödinger-Hirota equation. J. Nonlinear Opt. Phys. Mater. 23, 1450014 (2014)CrossRefGoogle Scholar
  20. 20.
    Biswas, A., Lott, D.A., Sutton, B., Khan, K.R., Mahmood, M.F.: Optical Gaussons in nonlinear directional couplers. J. Electromagn. Waves Appl. 27, 1976–1985 (2013)CrossRefGoogle Scholar
  21. 21.
    Banaja, M.A., Alkhateeb, S.A., Alshaery, A.A., Hilal, E.M., Bhrawy, A.H., Moraru, L., Biswas, A.: Optical solitons in dual-core couplers. Wulfenia 21, 366–381 (2014)Google Scholar
  22. 22.
    Rand, D., Glesk, I., Brès, C.S., Nolan, D.A., Chen, X., Koh, J., Fleischer, J.W., Steiglitz, K., Prucnal, P.R.: Observation of temporal vector soliton propagation and collision in birefringent fiber. Phys. Rev. Lett. 98, 053902 (2007) Google Scholar
  23. 23.
    Yang, J.K.: Interactions of vector solitons. Phys. Rev. E 64, 026607 (2001)CrossRefGoogle Scholar
  24. 24.
    Tang, D.Y., Man, W.S., Tam, H.Y., Drummond, P.D.: Observation of bound states of solitons in a passively mode-locked fiber laser. Phys. Rev. A 64, 033814 (2001)CrossRefGoogle Scholar
  25. 25.
    Tang, D.Y., Zhao, B., Shen, D.Y., Lu, C., Man, W.S., Tam, H.Y.: Bound-soliton fiber laser. Phys. Rev. A 66, 033806 (2002)CrossRefGoogle Scholar
  26. 26.
    Malomed, B.A.: Bound solitons in coupled nonlinear Schrödinger equations. Phys. Rev. A 45, R8321 (1992)CrossRefGoogle Scholar
  27. 27.
    Menyuk, C.R.: Pulse propagation in an elliptically birefringent Kerr medium. IEEE J. Quantum Electron. 25, 2674–2682 (1989)Google Scholar
  28. 28.
    Kaup, D.J., Malomed, B.A.: Soliton trapping and daughter waves in the Manakov model. Phys. Rev. A 48, 599 (1993)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Radhakrishnan, R., Lakshmanan, M., Hietarinta, J.: Inelastic collision and switching of coupled bright solitons in optical fibers. Phys. Rev. E 56, 2213–2216 (1997)CrossRefGoogle Scholar
  30. 30.
    Sun, Z.Y., Gao, Y.T., Yu, X., Liu, W.J., Liu, Y.: Bound vector solitons and soliton complexes for the coupled nonlinear Schrödinger equations. Phys. Rev. E 80, 066608 (2009)CrossRefGoogle Scholar
  31. 31.
    Kanna, T., Lakshmanan, M.: Exact soliton solutions, shape changing collisions, and partially coherent solitons in coupled nonlinear Schrödinger equations. Phys. Rev. Lett. 86, 5043–5046 (2001)CrossRefGoogle Scholar
  32. 32.
    Vijayajayanthi, M., Kanna, T., Lakshmanan, M.: Bright-dark solitons and their collisions in mixed N-coupled nonlinear Schrödinger equations. Phys. Rev. A 77, 013820 (2008)CrossRefGoogle Scholar
  33. 33.
    Radhakrishnan, R., Dinda, P.T., Millot, G.: Efficient control of the energy exchange due to the Manakov vector-soliton collision. Phys. Rev. E 69, 046607 (2004)CrossRefGoogle Scholar
  34. 34.
    Malomed, B.A.: Polarization dynamics and interactions of solitons in a birefringent optical fiber. Phys. Rev. A 43, 410–423 (1991)Google Scholar
  35. 35.
    Muraki, D.J., Kath, W.L.: Polarization dynamics for solitons in birefringent optical fibers. Phys. Lett. A 139, 379–383 (1989)CrossRefGoogle Scholar
  36. 36.
    Liu, W.J., Tian, B., Zhang, H.Q.: Types of solutions of the variable-coefficient nonlinear Schrödinger equation with symbolic computation. Phys. Rev. E 78, 066613 (2008)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Liu, W.J., Tian, B., Zhang, H.Q., Xu, T., Li, H.: Solitary wave pulses in optical fibers with normal dispersion and higher-order effects. Phys. Rev. A 79, 063810 (2009)CrossRefGoogle Scholar
  38. 38.
    Hirota, R.: Exact envelope-soliton solutions of a nonlinear wave equation. J. Math. Phys. 14, 805–809 (1973)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Podlipensky, A., Szarniak, P., Joly, N.Y., Poulton, C.G., Russel, PStJ: Bound soliton pairs in photonic crystal fiber. Opt. Express 15, 1653–1662 (2007)CrossRefGoogle Scholar
  40. 40.
    Hirota, R.: Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons. Phys. Rev. Lett. 27, 1192–1194 (1971)Google Scholar
  41. 41.
    Haelterman, M., Sheppard, A.P., Snyder, A.W.: Bound-vector solitary waves in isotropic nonlinear dispersive media. Opt. Lett. 18, 1406–1408 (1993)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Wen-Jun Liu
    • 1
  • Nan Pan
    • 1
  • Long-Gang Huang
    • 1
  • Ming Lei
    • 1
  1. 1.State Key Laboratory of Information Photonics and Optical CommunicationsSchool of Science, Beijing University of Posts and TelecommunicationsBeijingChina

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