Skip to main content
SpringerLink
Log in

Periodic attenuating oscillation between soliton interactions for higher-order variable coefficient nonlinear Schrödinger equation

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

Abstract

According to the change in the amplitude of the oscillation, it can be divided into equal-amplitude oscillation, amplitude-reduced oscillation (attenuating oscillation) and amplitude-increasing oscillation (divergence oscillation). In this paper, the periodic attenuating oscillation of solitons for a higher-order variable coefficient nonlinear Schrödinger equation is investigated. Analytic one- and two-soliton solutions of this equation are obtained by the Hirota bilinear method. By analyzing the soliton propagation properties, we study how to choose the corresponding parameters to control the soliton propagation and periodic attenuation oscillation phenomena. Results might be of significance for the study of optical communications including soliton control, amplification, compression and interactions.

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

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. 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 

  3. 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 

  4. 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 

  5. 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 

  6. 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 

  7. 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 

  8. 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 

  9. 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 

  10. 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 

  11. 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 

  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. 75, 3589–3603 (2018)

    Article  MathSciNet  Google Scholar 

  13. 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  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. 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 

  16. 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 

  17. 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 

  18. 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 

  19. 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 

  20. 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 

  21. Agrawal G.P.: Applications of Nonlinear Fiber Optics Academic. San Diego (2001)

  22. Goyal, A., Gupta, R., Kumar, C.N., Raju, T.S.: Chirped femtosecond solitons and double-kink solitons in the cubic-quintic nonlinear Schrödinger equation with self-steepening and self-frequency shift. Phys. Rev. A 84, 063830 (2011)

    Article  Google Scholar 

  23. Liu, W.J., Liu, M., Han, H., Fang, S., Teng, H., Lei, M., Wei, Z.: Nonlinear optical properties of WSe\(_{2}\) and MoSe\(_{2}\) films and their applications in passively Q-switched erbium doped fiber lasers. Photonics Res. 6, C15–C21 (2018)

    Article  Google Scholar 

  24. Liu, M.L., OuYang, Y.Y., Hou, H.R., Lei, M., Liu, W.J., Wei, Z.Y.: MoS\(_{2}\) saturable absorber prepared by chemical vapor deposition method for nonlinear control in Q-switching fiber laser. Chin. Phys. B. 27, 084211 (2018)

    Article  Google Scholar 

  25. Liu, W.J., Liu, M.L., OuYang, Y.Y., Hou, H.R., Lei, M., Wei, Z.: CVD-grown MoSe\(_{2}\) with high modulation depth for ultrafast mode-locked erbium-doped fiber laser. Nanotechnology 29, 394002 (2018)

    Article  Google Scholar 

  26. Zhang, Y., Yang, C., Yu, W., Liu, M., Ma, G., Liu, W.: Some types of dark soliton interactions in inhomogeneous optical fibers. Opt. Quant. Electron. 50, 295 (2018)

    Article  Google Scholar 

  27. Yu, W., Ekici, M., Mirzazadeh, M., Zhou, Q., Liu, W.: Periodic oscillations of dark solitons in nonlinear optics. Optik 165, 341–344 (2018)

    Article  Google Scholar 

  28. Yu, W., Yang, C., Liu, M., Zhang, Y., Liu, W.: Interactions of solitons, dromion-like structures and butterfly-shaped pulses for variable coefficient nonlinear Schrödinger equation. Optik 159, 21–30 (2018)

    Article  Google Scholar 

  29. Li, W., OuYang, Y.Y., Ma, G., Liu, M.L., Liu, W.J.: Q-switched all-fiber laser with short pulse duration based on tungsten diselenide. Laser Phys. 28(5), 055104 (2018)

    Article  Google Scholar 

  30. Liu, W., Liu, M., Yin, J., Chen, H., Lu, W., Fang, S., Teng, H., Lei, M., Yan, P., Wei, Z.: Tungsten diselenide for all-fiber lasers with the chemical vapor deposition method. Nanoscale 10, 7971–7977 (2018)

    Article  Google Scholar 

  31. Liu, M., Liu, W., Yan, P., Fang, S., Teng, H., Wei, Z.: High-power MoTe\(_{2}\)-based passively Q-switched erbium-doped fiber laser. Chin. Opt. Lett. 16, 020007 (2018)

    Article  Google Scholar 

  32. Xu, Z.Y., Li, L., Li, Z.H., Zhou, G.S.: Modulation instability and solitons on a cw background in an optical fiber with higher-order effects. Phys. Rev. E 67, 026603 (2003)

    Article  MathSciNet  Google Scholar 

  33. Kumar, C.N., Durganandini, P.: New phase modulated solution of higher-order nonlinear Schröodinger equation. Pramana J. Phys. 53, 271–277 (1999)

    Google Scholar 

  34. Raza Rizvi, S.T., Ali, K., Akram, U.: Analytical study of solitons for Lakshmanan–Porsezian–Daniel model with parabolic law nonlinearity. Optik 168, 27–33 (2018)

    Article  Google Scholar 

  35. Ali, K., Rizvi, S.T.R., Ahmad, S., Bashir, S., Younis, M.: Bell and kink type soliton solutions in birefringent nano-fibers. Optik 142, 327–333 (2017)

    Article  Google Scholar 

  36. Rizvi, S.T.R., Ali, K.: Jacobian elliptic periodic traveling wave solutions in the negative-index materials. Nonlinear Dyn. 87(3), 1967–1972 (2017)

    Article  Google Scholar 

  37. 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: CaF\(_{2}\) laser near 2 \(\mu \)m. Opt. Laser Technol. 103, 89–92 (2018)

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  41. 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 

  42. 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: SrF\(_{2}\) ceramic laser. Opt. Eng. 55(10), 106114 (2016)

    Article  Google Scholar 

  43. 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: CaF\(_{2}\) disordered crystal. Opt. Laser Technol. 69, 140–143 (2015)

    Article  Google Scholar 

  44. 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 

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

    Article  Google Scholar 

  46. 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 

  47. 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:Sc\(_{2}\)SiO\(_{5}\) mode-locked laser. Opt. Laser Technol. 65, 1–4 (2015)

    Article  Google Scholar 

  48. 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: CaF\(_{2}\) laser mode-locked by monolayer graphene. Opt. Laser Technol. 75, 83–86 (2015)

    Article  Google Scholar 

  49. Palacios, S.L., Fernández-Díaz, J.M.: Black optical solitons for media with parabolic nonlinearity law in the presence of fourth order dispersion. Opt. Commun. 178, 457–460 (2000)

    Article  Google Scholar 

  50. Akhmediev, N.N., Ankiewicz, A.: Solitons: Nonlinear Pulses and Beams. Chapman and Hall, London (1997)

    MATH  Google Scholar 

  51. Porsezian, K., Nakkeeran, K.: Optical solitons in presence of Kerr dispersion and self-frequency shift. Phys. Rev. Lett. 76, 3955 (1996)

    Article  Google Scholar 

  52. Ankiewicz, A., Akhmediev, N.: Moving fronts for complex Ginzburg–Landau equation with Raman term. Phys. Rev. E. 58, 6723–6727 (1998)

    Article  Google Scholar 

  53. Shen, S., Chang, C.C., Sardesai, H.P., Binjrajka, V., Weiner, A.M.: Effects of self-phase modulation on sub-500 fs pulse transmission over dispersion compensated fiber links. IEEE J. Lightwave Technol. 17, 452–461 (1999)

    Article  Google Scholar 

  54. Raghavan, S., Agrawal, G.P.: Spatiotemporal solitons in inhomogeneous nonlinear media. Opt. Commun. 180, 377–382 (2000)

    Article  Google Scholar 

  55. Hasegawa, A., Tappert, F.: Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion. Appl. Phys. Lett. 23, 171 (1973)

    Article  Google Scholar 

  56. Mollenauer, L.F., Stolen, R.H., Gordon, J.P.: Experimental observation of picosecond pulse narrowing and solitons in optical fibers. Phys. Rev. Lett. 45, 1095 (1980)

    Article  Google Scholar 

  57. Agrawal, G.P.: Modulation instability induced by cross-phase modulation. Phys. Rev. Lett. 59, 880 (1987)

    Article  Google Scholar 

  58. Choudhuri, A., Triki, H., Porsezian, K.: Self-similar localized pulses for the nonlinear Schrödinger equation with distributed cubic-quintic nonlinearity. Phys. Rev. A 94, 063814 (2016)

    Article  Google Scholar 

  59. He, J., Zhang, J., Zhang, M.Y., Dai, C.Q.: Analytical nonautonomous soliton solutions for the cubic quintic nonlinear Schrödinger equation with distributed coefficients. Opt. Commun. 285, 755 (2012)

    Article  Google Scholar 

  60. Yang, R.C., Li, L., Hao, R.Y., Li, Z.H., Zhou, G.S.: Combined solitary wave solutions for the inhomogeneous higher-order nonlinear Schrödinger equation. Phys. Rev. E 71, 036616 (2005)

    Article  MathSciNet  Google Scholar 

  61. Soloman Raju, T., Panigrahi, P.K., Porsezian, K.: Nonlinear compression of solitary waves in asymmetric twin-core fibers. Phys. Rev. E 71, 026608 (2005)

    Article  Google Scholar 

  62. Ponomarenko, S.A., Agrawal, G.P.: Do solitonlike self-similar waves exist in nonlinear optical media. Phys. Rev. Lett. 97, 013901 (2006)

    Article  Google Scholar 

  63. Tao, Y., He, J.: Multisolitons, breathers, and rogue waves for the Hirota equation generated by the Darboux transformation. Phys. Rev. E 85, 026601 (2012)

    Article  Google Scholar 

  64. Kodama, Y., Hasegawa, A.: Nonlinear pulse propagation in a monomode dielectric guide. IEEE J. Quantum Electron. 23, 510 (1987)

    Article  Google Scholar 

  65. Yang, G., Li, L., Jia, S.: Peregrine rogue waves induced by the interaction between a continuous wave and a soliton. Phys. Rev. E 85, 046608 (2012)

    Article  Google Scholar 

  66. Hirota, R.: Exact envelope-soliton solutions of a nonlinear wave equation. J. Math. Phys. 14, 805 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  67. Sun, W.R.: Breather-to-soliton transitions and nonlinear wave interactions for the nonlinear Schrödinger equation with the sextic operators in optical fibers. Ann. Phys. 529, 1600227 (2017)

    Article  MATH  Google Scholar 

  68. Wang, L., Zhang, J.H., Wang, Z.Q., Liu, C., Li, M., Qi, F., Guo, R.: Breather transition dynamics, Peregrine combs and walls, and modulation instability in a variable-coefficient nonlinear Schrödinger equation with higher-order effects. Phys. Rev. E 93, 012214 (2016)

    Article  MathSciNet  Google Scholar 

  69. Chowdury, A., Kedziora, D.J., Ankiewicz, A., Akhmediev, N.: Breather-to-soliton conversions described by the quintic equation of the nonlinear Schrödinger hierarchy. Phys. Rev. E 91, 032928 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  70. Chowdury, A., Ankiewicz, A., Akhmediev, N.: Moving breathers and breather-to-soliton conversions for the Hirota equation. Proc. R. Soc. A 471, 20150130 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  71. Zhou, S., Kuznetsova, L., Chong, A., Wise, F.W.: Compensation of nonlinear phase shifts with third-order dispersion: fiber stretchers can out-perform grating stretchers in short-pulse fiber amplifiers. Opt. Express 13(13), 4869–77 (2005)

    Article  Google Scholar 

  72. Essiambre, R.J., Raybon, G., Mikkelsen, B.: Pseudo-linear transmission of high-speed TDM signals: 40 and 160 Gb/s. In: Kaminow, I., Li, T. (eds.) Optical Fiber Telecommunications IV B, Systems and Impairments. Academic Press, San Diego (2002)

    Google Scholar 

  73. Ablowitz, M.J., Clarkson, P.A.: Solitons, Nonlinear Evolution Equations and Scattering. Cambridge University Press, Cambridge (1992)

    MATH  Google Scholar 

  74. Segev, M., Stegeman, G.: Self-trapping of optical beams: spatial solitons. Phys. Today 51, 42 (1998)

    Article  Google Scholar 

  75. Liu, W.J., Tian, B., Lei, M.: Elastic and inelastic interactions between optical spatial solitons in nonlinear optics. Laser. Phys. 23, 095401 (2013)

    Article  Google Scholar 

  76. Pinto, A.N., Agrawal, G.P.: Nonlinear interaction between signal and noise in optical fibers. J. Lighwave Technol. 26, 1847–1853 (2008)

    Article  Google Scholar 

  77. Xie, C., Karlsson, M., Andrekson, P.A.: Influences of polarization-mode dispersion on soliton transmission systems. IEEE J. Sel. Top. Quantum Electron. 8, 575 (2002)

    Article  Google Scholar 

  78. Ganapathy, R., Porsezian, K., Hasegawa, A.: Soliton interaction under soliton dispersion management. IEEE. J. Quantum Electron. 44, 383–390 (2008)

    Article  Google Scholar 

  79. Desem, C., Chu, P.L.: Soliton interaction in the presence of loss and periodic amplification in optical fibers. Opt. Lett. 12, 349–351 (1987)

    Article  Google Scholar 

  80. Kodama, Y., Nozaki, K.: Soliton interaction in optical fibers. Opt. Lett. 12, 1038–1040 (1987)

    Article  Google Scholar 

  81. Peng, G.D., Ankiewicz, A.: Fundamental and second order soliton transmission in nonlinear directional fiber couplers. J. Nonlinear Opt. Phys. 1, 135 (1992)

    Article  Google Scholar 

  82. Friberg, S.R.: Demonstration of colliding-soliton all-optical switching. Appl. Phys. Lett. 63, 429–431 (1993)

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  84. Nimmo, J.J.C., Freeman, N.C.: A method of obtaining the N-soliton solution of the Boussinesq equation in terms of a Wronskian. J. Phys. A 17, 1415 (1984)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

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 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, China

    Xiaoyan Liu & 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. Xiaoyan Liu
    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 authors

Correspondence to Wenjun Liu or Qin Zhou.

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

Liu, X., Liu, W., Triki, H. et al. Periodic attenuating oscillation between soliton interactions for higher-order variable coefficient nonlinear Schrödinger equation. Nonlinear Dyn 96, 801–809 (2019). https://doi.org/10.1007/s11071-019-04822-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-019-04822-z

Keywords

Search

Navigation