Skip to main content
SpringerLink
Log in

Dynamic behaviors of general N-solitons for the nonlocal generalized nonlinear Schrödinger equation

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

Abstract

The general N-solitons of nonlocal generalized nonlinear Schrödinger equations with third-order, fourth-order and fifth-order dispersion terms and nonlinear terms (NGNLS) are studied. Firstly, the Riemann–Hilbert problem and the general N-soliton solutions of NGNLS equations were given. Then, we study the symmetry relations of the eigenvalues and eigenvectors related to the scattering data which involve the reverse-space, reverse-time and reverse-space-time reductions. Thirdly, some novel solitons and the dynamic behaviors which corresponded to novel eigenvalue configurations and the coefficients of higher-order terms are given. In all the three NGNLS equations, their solutions often collapse periodically, but can remain bounded or nonsingular for wide ranges of soliton parameters as well. In addition, it is found that the higher-order terms of the NGNLS equations not only affect the amplitude variation of the soliton, but also influence the singularity and the motion of the soliton.

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.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. Gardner, C.S., Greene, J.M., Kruskal, M.D., Miura, R.M.: Nonlinear-evolution equations of physical significance. Phys. Rev. Lett. 31(2), 125–127 (1973)

    Article  MathSciNet  Google Scholar 

  2. Zakharov, V.E., Shabat, A.B.: Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. J. Exp. Theor. Phys. 34, 62–69 (1972)

    MathSciNet  Google Scholar 

  3. Ablowitz, M.J., Kaup, D.J., Newell, A.C., Segur, H.: The inverse scattering transform-Fourier analysis for nonlinear problems. Stud. Appl. Math. 53(4), 249–315 (1974)

    Article  MathSciNet  Google Scholar 

  4. Ablowitz, M.J., Clarkson, P.A.: Solitons. Cambridge University Press, Cambridge, Nonlinear evolution equations and inverse scattering (1991)

    MATH  Google Scholar 

  5. Ablowitz, M.J., Fokas, A.S.: Complex variables, introduction and applications. Cambridge University Press, Cambridge (2003)

    Book  Google Scholar 

  6. Novikov, S., Manakov, S. V., Pitaevskii, L. P., Zakharov, V. E.: Theory of Solitons:the Inverse Scattering Method. (1984)

  7. Bender, M.C., Boettcher, S.: Real spectra in non-Hermitian Hamiltonians having PT symmetry. Phys. Rev. Lett. 80, 5243 (1998)

    Article  MathSciNet  Google Scholar 

  8. Ablowitz, M.J., Musslimani, Z.H.: Integrable nonlocal nonlinear Schrödinger equation. Phys. Rev. Lett. 110, 064105 (2013)

    Article  Google Scholar 

  9. Ablowitz, M.J., Musslimani, Z.H.: Inverse scattering transform for the integrable nonlocal nonlinear Schrödinger equation. Nonlinearity 29, 915–946 (2016)

    Article  MathSciNet  Google Scholar 

  10. Yang, J. K.: Nonlinear waves in integrable and nonintegrable systems. Society for Industrial and Applied Mathematics. (2010)

  11. Guo, B.L., Liu, N., Wang, Y.: A Riemann-Hilbert approach for a new type coupled nonlinear Schrödinger equations. J. Math. Anal. Appl. 459(1), 145–158 (2017)

    Article  Google Scholar 

  12. Peng, W.Q., Tian, S.F., Wang, X.B., et al.: Riemann-Hilbert method and multi-soliton solutions for three-component coupled nonlinear Schrödinger equations. J. Geom. Phys. 146, 103508 (2019)

    Article  MathSciNet  Google Scholar 

  13. Wang, J., Su, T., Geng, X.G.: Riemann-Hilbert approach and N -soliton solutions for a new two-component Sasa-Satsuma equation. Nonlinear Dynam. 101(1), 597–609 (2020)

    Article  Google Scholar 

  14. Zhao, Y., Fan, E.G.: \(N\)-soliton solution for a higher-order Chen-Lee-Liu equation with nonzero boundary conditions. Mod. Phys. Lett. B 34(04), 339 (2020)

    Article  MathSciNet  Google Scholar 

  15. Li, M., Xu, T.: Dark and antidark soliton interactions in the nonlocal nonlinear Schrödinger equation with the self-induced parity-time-symmetric potential. Phys. Rev. E 91, 033202 (2015)

    Article  MathSciNet  Google Scholar 

  16. Yan, Z.Y.: Integrable PT-symmetric local and nonlocal vector nonlinear Schrödinger equations: A unified two-parameter model. Appl. Math. Lett. 47, 61–68 (2015)

    Article  MathSciNet  Google Scholar 

  17. Yan, Z.Y., Chen, Y., Wen, Z.C.: On stable solitons and interactions of the generalized Gross-Pitaevskii equation with PT-, and non-PT-symmetric potentials. Chaos 26(8), 5243 (2016)

    Article  MathSciNet  Google Scholar 

  18. Huang, X., Ling, L.M.: Soliton solutions for the nonlocal nonlinear Schrödinger equation. Eur. Phys. J. Plus 131(5), 1–11 (2016)

    Article  Google Scholar 

  19. Wen, X.Y., Yan, Z.Y., Yang, Y.Q.: Dynamics of higer-order rational solitons for the nonlocal nonlinear Schrödinger equation with the self-induced parity-time-symmetric potential. Chaos 26, 063123 (2016)

    Article  MathSciNet  Google Scholar 

  20. Ji, J.L., Zhu, Z.N.: Soliton solutions of an integrable nonlocal modified Korteweg-de Vries equation through inverse scattering transform. J. Math. Anal. Appl. 453(2), 973–984 (2017)

    Article  MathSciNet  Google Scholar 

  21. Zhang, H.Q., Tian, B., Li, L.L.: Darboux transformation and soliton solutions for the (2+1)-dimensional nonlinear Schrödinger hierarchy with symbolic computation. Physica A 388(1), 9–20 (2009)

    Article  MathSciNet  Google Scholar 

  22. Yang, B., Chen, Y.: Dynamics of rogue waves in the partially pt-symmetric nonlocal Davey-Stewartson systems. Commun. Nonl. Sci. Numer. Simulat. 69, 287–303 (2019)

    Article  MathSciNet  Google Scholar 

  23. Yang, J. K.: General \(N\)-solitons and their dynamics in several nonlocal nonlinear Schrödinger equations. Phys. Lett. A 282 (4) (2019)

  24. Lakshmanan, M., Porsezian, K., Daniel, M.: Effect of discreteness on the continuum limit of the Heisenberg spin chain. Phys. Lett. A 133(9), 483–488 (1988)

    Article  Google Scholar 

  25. Christov, I.P.: Enhanced generation of attosecond pulses in dispersion-controlled hollow-core fiber. Phys. Rev. A 60(4), 3244–3250 (1999)

    Article  Google Scholar 

  26. Zhang, X.E., Chen, Y.: Inverse scattering transformation for generalized nonlinear Schrödinger equation. Appl. Math. Lett. 98, 306–313 (2019)

    Article  MathSciNet  Google Scholar 

  27. Xu, T., He, G.L.: Higher-order interactional solutions and rogue wave pairs for the coupled Lakshmanan-Porsezian-Daniel equations. Nonlinear. Dyna. 98(3), 1731–1744 (2019)

    Article  MathSciNet  Google Scholar 

  28. Chowdury, A., Kedziora, D. J., Ankiewicz, A., et al: Soliton solutions of an integrable nonlinear Schrödinger equation with quintic terms. Phys. Rev. E 90(3), (2014)

  29. Zakharov, V.E., Shabat, A.B.: Integration of the nonlinear equations of mathematical physics by the method of the inverse scattering problem II. Funct. Anal. Appl. 13, 166–174 (1979)

  30. Faddeev, L.D., Takhtadjan, L.A.: Hamiltonian methods in the theory of solitons. Springer, Berlin (1987)

    Book  Google Scholar 

  31. Nimmo, J.J.C.: A bilinear B\(\ddot{a} \)cklund transformation for the nonlinear Schrödinger equation. Phys. Lett. A. 99, 279–280 (1983)

    Article  MathSciNet  Google Scholar 

  32. Yang, Y.Q., Yan, Z.Y., Malomed, B.A.: Rogue waves, rational solitons and modulational instability in an integrable fifth-order nonlinear Schrödinger equation. Chaos 25, 1–11 (2015)

    MATH  Google Scholar 

  33. Lan, Z.Z., Gao, Y.T., Zhao, C., et al.: Dark soliton interactions for a fifth-order nonlinear Schrodinger equation in a Heisenberg ferromagnetic spin chain. Super. Micro. 100, 191–197 (2016)

    Article  Google Scholar 

  34. Yomba, E., Zakeri, G.A.: Collision of \(N\)-solitons in a fifth-order nonlinear Schrodinger equation. Wave Motion 72, 101–112 (2017)

    Article  MathSciNet  Google Scholar 

  35. Yang, Y.Q., Wang, X., Cheng, X.P.: Higher-order rational solutions for a new integrable nonlocal fifth-order nonlinear Schrödinger equation. Wave Motion 77, 1–11 (2018)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors gratefully acknowledge the support of National Natural Science Foundation of China (No. 11675054), Science and Technology Commission of Shanghai Municipality (No. 18dz2271000) and Future Scientist and Outstanding Scholar Cultivation Program of East China Normal University (No.WLKXJ202001).

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Shanghai Key Laboratory of PMMP, Shanghai Key Laboratory of PMMP, Shanghai Key Laboratory of Trustworthy Computing, East China Normal University, Shanghai, 200062, P. R. China

    Minmin Wang & Yong Chen

  2. College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, 266590, P. R. China

    Yong Chen

  3. Department of Physics, Zhejiang Normal University, Jinhua, 321004, P. R. China

    Yong Chen

Authors
  1. Minmin Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yong Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yong Chen.

Ethics declarations

Conflict of interest

The authors declare that there is no conflict of interests regarding the publication of this paper.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The project is supported by National Natural Science Foundation of China (No. 11675054), Science and Technology Commission of Shanghai Municipality (No. 18dz2271000) and Future Scientist and Outstanding Scholar Cultivation Program of East China Normal University (No.WLKXJ202001)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, M., Chen, Y. Dynamic behaviors of general N-solitons for the nonlocal generalized nonlinear Schrödinger equation. Nonlinear Dyn 104, 2621–2638 (2021). https://doi.org/10.1007/s11071-021-06421-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-021-06421-3

Keywords

Search

Navigation