Advertisement

Nonlinear Dynamics

, Volume 95, Issue 2, pp 983–994 | Cite as

Bright soliton interactions in a \(\mathbf (2 +\mathbf 1) \)-dimensional fourth-order variable-coefficient nonlinear Schrödinger equation for the Heisenberg ferromagnetic spin chain

  • Chunyu Yang
  • Qin ZhouEmail author
  • Houria Triki
  • Mohammad Mirzazadeh
  • Mehmet Ekici
  • Wen-Jun LiuEmail author
  • Anjan Biswas
  • Milivoj Belic
Original Paper

Abstract

Interactions of bright solitons in the Heisenberg ferromagnetic spin chain, governed by a \((2+1)\)-dimensional nonlinear Schrödinger equation with variable coefficients, are investigated theoretically. Analytical soliton solutions are derived by means of the Hirota bilinear method. Different scenarios of soliton propagation and interactions are illustrated. The influence of relevant parameters and variable coefficients of different function types on the soliton propagation and interactions is discussed. Solitons of different shapes in propagation, whose pulse widths and oscillation periods vary with the phase shift, are presented. The method for controlling the size of the phase shift is proposed. Furthermore, the fission solitons and the bound solitons, produced after their collisions, are displayed. Methods for controlling the number and amplitudes of fission solitons, the distance between solitons and the intensity of their interactions are put forward. Results in this paper will contribute to the effective control of solitons in the Heisenberg ferromagnetic spin chain system.

Keywords

Variable-coefficient nonlinear Schrödinger equation Heisenberg ferromagnetic spin chain Soliton propagation Soliton interaction 

Notes

Acknowledgements

The work of Qin Zhou was supported by the National Natural Science Foundation of China (Grant Nos. 11705130 and 1157149) and by the Chutian Scholar Program of Hubei Government in China. The work of Wenjun Liu was supported by 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 Nos. IPOC2016ZT04 and IPOC2017ZZ05). The research work of Milivoj Belic was supported by the Qatar National Research Fund (QNRF) under the Grant Number NPRP 8-028-1-001.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. 1.
    Wazwaz, A.M., El-Tantawy, S.A.: New \((3+1)\)-dimensional equations of Burgers type and Sharma–Tasso–Olver type: multiple-soliton solutions. Nonlinear Dyn. 87(4), 2457–2461 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Wazwaz, A.M.: Two-mode fifth-order KdV equations: necessary conditions for multiple-soliton solutions to exist. Nonlinear Dyn. 87(3), 1685–1691 (2017)MathSciNetCrossRefGoogle Scholar
  3. 3.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Wazwaz, A.M., El-Tantawy, S.A.: A new \((3+1)\)-dimensional generalized Kadomtsev–Petviashvili equation. Nonlinear Dyn. 84(2), 1107–1112 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Osman, M.S., Wazwaz, A.M.: An efficient algorithm to construct multi-soliton rational solutions of the \((2+1)\)-dimensional KdV equation with variable coefficients. Appl. Math. Comput. 321, 282–289 (2018)MathSciNetGoogle Scholar
  6. 6.
    Shukla, P.K., Eliasson, B.: Nonlinear aspects of quantum plasma physics. Phys. Usp. 53, 51–76 (2010)CrossRefGoogle Scholar
  7. 7.
    Busch, T., Anglin, J.R.: Dark-bright solitons in inhomogeneous Bose–Einstein condensates. Phys. Rev. Lett. 87, 010401 (2001)CrossRefGoogle Scholar
  8. 8.
    Sklarz, S.E., Tannor, D.J.: Loading a Bose–Einstein condensate onto an optical lattice: an application of optimal control theory to the nonlinear Schrödinger equation. Phys. Rev. A 66, 053619 (2002)CrossRefGoogle Scholar
  9. 9.
    Kibler, B., Fatome, J., Finot, C., Millot, G., Dias, F., Genty, G., Akhmediev, N., Dudley, J.M.: Observation of Kuznetsov-Ma soliton dynamics in optical fibre. Sci. Rep. 6, 463 (2012)CrossRefGoogle Scholar
  10. 10.
    Radhakrishnan, R., Kundu, A., Lakshmanan, M.: Coupled nonlinear Schrödinger equations with cubic-quintic nonlinearity: integrability and soliton interaction in non-Kerr media. Phys. Rev. E. 60, 3314–3323 (1999)CrossRefGoogle Scholar
  11. 11.
    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 (2018)CrossRefGoogle Scholar
  12. 12.
    Biswas, A., Zhou, M.Z., Ullah, M.Z., Triki, H., Moshokoa, S.P., Belic, M.: Optical soliton perturbation with anti-cubic nonlinearity by semi-inverse variational principle. Optik 143, 131–134 (2017)CrossRefGoogle Scholar
  13. 13.
    Chakraborty, S., Nandy, S., Barthakur, A.: Bilinearization of the generalized coupled nonlinear Schodinger equation with variable coefficients and gain and dark-bright pair soliton solutions. Phys. Rev. E 91, 023210 (2015)MathSciNetCrossRefGoogle Scholar
  14. 14.
    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 MoS\(_{2}\)-Sb\(_{2}\)Te\(_{3}\)-MoS\(_{2}\) heterostructure materials. Photonics Res. 6, 220–227 (2018)CrossRefGoogle Scholar
  15. 15.
    Li, W.Y., Ma, G.L., Yu, W.D., 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. 27, 030504 (2018)CrossRefGoogle Scholar
  16. 16.
    Biswas, A., Zhou, Q., Ullah, M.Z., Asma, M., Moshokoa, S.P., Belic, M.: Perturbation theory and optical soliton cooling with anti-cubic nonlinearity. Optik 142, 73–76 (2017)CrossRefGoogle Scholar
  17. 17.
    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. Sel. Top. Quant. 24, 0901005 (2018)Google Scholar
  18. 18.
    Kanna, T., Lakshmanan, M.: Exact soliton solutions, shape changing collisions, and partially coherent solitons in coupled nonlinear Schrödinger equations. Phys. Rev. Lett. 86, 5043 (2001)CrossRefGoogle Scholar
  19. 19.
    Biswas, A., Triki, H., Zhou, Q., Moshokoa, S.P., Ullah, M.Z., Belic, M.: Cubic-quartic optical solitons in Kerr and power law media. Optik 144, 357–362 (2017)CrossRefGoogle Scholar
  20. 20.
    Wong, P., Liu, W.J., Huang, L.G., Li, Y.Q., Pan, N., Lei, M.: Higher-order-effects management of soliton interactions in the Hirota equation. Phys. Rev. E 91, 033201 (2015)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Vladimir, N.S., Akira, H.: Novel soliton solutions of the nonlinear Schrödinger equation model. Phys. Rev. Lett. 85, 4502 (2000)CrossRefGoogle Scholar
  22. 22.
    Biswas, A., Zhou, Q., Moshokoa, S.P., Triki, H., Belic, M., Alqahtani, R.T.: Resonant 1-soliton solution in anti-cubic nonlinear medium with perturbations. Optik 45, 14–17 (2017)CrossRefGoogle Scholar
  23. 23.
    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)CrossRefzbMATHGoogle Scholar
  24. 24.
    Loomaba, S., Pal, R., Kumar, C.N.: Bright solitons of the nonautonomous cubic-quintic nonlinear Schodinger equation with sign-reversal nonlinearity. Phys. Rev. A 92, 033811 (2015)CrossRefGoogle Scholar
  25. 25.
    Biswas, A., Ullah, M.Z., Zhou, Q., Moshkoa, S.P., Triki, H., Belic, M.: Resonant optical solitons with quadratic-cubic nonlinearity by semi-inverse variational principle. Optik 145, 18–21 (2017)CrossRefGoogle Scholar
  26. 26.
    Chowdury, A., Kedziora, D.J., Ankiewicz, A., Akhmediev, N.: Soliton solutions of an integrable nonlinear Schrödinger equation with quintic terms. Phys. Rev. E 90, 032922 (2014)CrossRefGoogle Scholar
  27. 27.
    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(4), 042201 (2017)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Biswas, A., Kara, A.H., Ullah, M.Z., Zhou, Q., Triki, H., Belic, M.: Conservation laws for cubic-quartic optical solitons in Kerr and power law media. Optik 145, 650–654 (2017)CrossRefGoogle Scholar
  29. 29.
    Serak, S.V., Tabiryan, N.V., Peccianti, M., Assanto, G.: Spatial soliton all-optical logic gates. IEEE Photonics Technol. Lett. 18, 1287–1289 (2006)CrossRefGoogle Scholar
  30. 30.
    Wu, Y.: New all-optical switch based on the spatial soliton repulsion. Opt. Express 14, 4005–4012 (2006)CrossRefGoogle Scholar
  31. 31.
    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 (2018)CrossRefGoogle Scholar
  32. 32.
    Shi, T.T., Chi, S.: Nonlinear photonic switching by using the spatial soliton collision. Opt. Lett. 15, 1123–1125 (1990)CrossRefGoogle Scholar
  33. 33.
    Tanguy, Y., Ackemann, T., Firth, W.J., Jäger, R.: Realization of a semiconductor-based cavity soliton laser. Phys. Rev. Lett. 100, 013907 (2008)CrossRefGoogle Scholar
  34. 34.
    Li, W.Y., OuYang, Y.Y., Ma, G.L., Liu, M.L., Liu, W.J.: Q-switched all-fiber laser with short pulse duration based on tungsten diselenide. Laser Phys. 28, 055104 (2018)CrossRefGoogle Scholar
  35. 35.
    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 (2018)CrossRefGoogle Scholar
  36. 36.
    Liu, D.Y., Tian, B., Jiang, Y., Xie, X.Y., Wu, X.Y.: Analytic study on a \((2+1)\)-dimensional nonlinear Schrödinger equation in the Heisenberg ferromagnetism. Comput. Math. Appl. 71, 2001–2007 (2016)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Tang, G.S., Wang, S.H., Wang, G.W.: Solitons and complexitons solutions of an integrable model of \((2+1)\)-dimensional Heisenberg ferromagnetic spin chain. Nonlinear Dyn. 88, 2319–2327 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Wang, Q.M., Gao, Y.T., Su, C.Q., Mao, B.Q., Gao, Z., Yang, J.W.: Dark solitonic interaction and conservation laws for a higher-order \((2+1)\)-dimensional nonlinear Schrödinger-type equation in a Heisenberg ferromagnetic spin chain with bilinear and biquadratic interaction. Ann. Phys. 363, 440–456 (2015)CrossRefzbMATHGoogle Scholar
  39. 39.
    Inc, M., Aliyu, A.I., Yusuf, A., Baleanu, D.: Optical solitons and modulation instability analysis of an integrable model of \((2+1)\)-dimensional Heisenberg ferromagnetic spin chain equation. Superlattice Microstruct. 112, 628–638 (2017)CrossRefGoogle Scholar
  40. 40.
    Trikia, H., Wazwaz, A.M.: New solitons and periodic wave solutions for the \((2+1)\)-dimensional Heisenberg ferromagnetic spin chain equation. J. Electromagn. Waves Appl. 30, 788–794 (2016)CrossRefGoogle Scholar
  41. 41.
    Latha, M.M., Vasanthi, C.C.: An integrable model of \((2+1)\)-dimensional Heisenberg ferromagnetic spin chain and soliton excitations. Phys. Scr. 89, 065204 (2014)CrossRefGoogle Scholar
  42. 42.
    Lan, Z.Z., Gao, B.: Lax pair, infinitely many conservation laws and solitons for a \((2+1)\)-dimensional Heisenberg ferromagnetic spin chain equation with time-dependent coefficients. Appl. Math. Lett. 79, 6–12 (2018)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Zhao, X.H., Tian, B., Liu, D.Y., Wu, X.Y., Chai, J., Guo, Y.J.: Dark solitons interaction for a \((2+1)\)-dimensional nonlinear Schrödinger equation in the Heisenberg ferromagnetic spin chain. Superlattice Microstruct. 100, 587–595 (2016)CrossRefGoogle Scholar
  44. 44.
    Vasanthi, C.C., Latha, M.M.: Heisenberg ferromagnetic spin chain with bilinear and biquadratic interactions in \((2+1)\) dimensions. Commun. Nonlinear Sci. Numer. Simul. 28, 109–122 (2015)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Xie, X.Y., Tian, B., Chai, J., Wu, X.Y., Jiang, Y.: Dark soliton collisions for a fourth-order variable-coefficient nonlinear Schö dinger equation in an inhomogeneous Heisenberg ferromagnetic spin chain or alpha helical protein. Nonlinear Dyn. 86(1), 131–135 (2016)CrossRefGoogle Scholar
  46. 46.
    Yang, J.W., Gao, Y.T., Wang, Q.M., Su, C.Q., Feng, Y.J., Yu, X.: Bilinear forms and soliton solutions for a fourth-order variable-coefficient nonlinear Schrödinger equation in an inhomogeneous Heisenberg ferromagnetic spin chain or an alpha helical protein. Physica B 481, 148–155 (2016)CrossRefGoogle Scholar
  47. 47.
    Zhao, X.H., Tian, B., Liu, D.Y., Wu, X.Y.: Dark solitons, Lax pair and infinitely-many conservation laws for a generalized \((2+1)\)-dimensional variable-coefficient nonlinear Schröodinger equation in the inhomogeneous Heisenberg ferromagnetic spin chain. Mod. Phys. Lett. B 31, 1750013 (2017)CrossRefGoogle Scholar
  48. 48.
    Hirota, R.: Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons. Phys. Rev. Lett. 27, 1192–1194 (1971)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

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.School of Electronics and Information EngineeringWuhan Donghu UniversityWuhanPeople’s Republic of China
  3. 3.Radiation Physics Laboratory, Department of Physics, Faculty of SciencesBadji Mokhtar UniversityAnnabaAlgeria
  4. 4.Department of Engineering Sciences, Faculty of Technology and Engineering, East of GuilanUniversity of GuilanRudsar-VajargahIran
  5. 5.Department of Mathematics, Faculty of Science and ArtsBozok UniversityYozgatTurkey
  6. 6.Department of Physics, Chemistry and MathematicsAlabama A&M UniversityNormalUSA
  7. 7.Department of Mathematics and StatisticsTshwane University of TechnologyPretoriaSouth Africa
  8. 8.Science ProgramTexas A&M University at QatarDohaQatar

Personalised recommendations