Skip to main content
SpringerLink
Log in

Volumes of second-order rogue waves of the infinite NLS hierarchy

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

Abstract

We analyze first- and second-order rogue waves of the equations in the nonlinear Schrödinger equation (NLS) hierarchy. A physical phenomenon may be described by an individual equation or by a combination of them. We focus on localized (in 2D) formations. Then we can find the ‘volumes’ of these rogue waves and classify the formations as rogue or ‘semi-rogue’ waves. In other cases, there can be a rogue-type central structure connected to ‘soliton tails,’ and, in these cases, the standard ‘volume’ may be infinite, and a ’modified volume’ can be introduced.

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
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

Data availability

No data is needed.

References

  1. Ankiewicz, A., Akhmediev, N.: Multi-rogue waves and triangular numbers. Rom. Rep. Phys. 69, 104 (2017)

    Google Scholar 

  2. Ding, C.C., Gao, Y.T., Li, L.Q.: Breathers and rogue waves on the periodic background for the Gerdjikov–Ivanov equation for the Alfvén waves in an astrophysical plasma. Chaos, Solitons Fractals 120, 259–265 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  3. Kedziora, D.J., Ankiewicz, A., Akhmediev, N.: Rogue waves and solitons on a cnoidal background. Eur. Phys. J. Spec. Top. 223, 43 (2014)

    Article  Google Scholar 

  4. Ding, C.C., Gao, Y.T., Xin, Y., Liu, F.Y., Xi-Hu, W.: N-fold generalized Darboux transformation and breather–rogue waves on the constant/periodic background for a generalized mixed nonlinear Schrödinger equation. Nonlinear Dyn. 109, 989–1004 (2022). https://doi.org/10.1007/s11071-022-07423-5

    Article  Google Scholar 

  5. Kundu, A.: Landau–Lifshitz and higher-order nonlinear systems gauge generated from nonlinear Schrödinger-type equations. J. Math. Phys. 25, 3433–3438 (1984). https://doi.org/10.1063/1.526113

    Article  ADS  MathSciNet  Google Scholar 

  6. Ankiewicz, A., Kedziora, D.J., Chowdury, A., Bandelow, U., Akhmediev, N.: Infinite hierarchy of nonlinear Schrödinger equations and their solutions. Phys. Rev. E 93, 012206 (2016)

    Article  ADS  MathSciNet  CAS  PubMed  Google Scholar 

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

    Google Scholar 

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

    Article  ADS  Google Scholar 

  9. Porsezian, K., Daniel, M., Lakshmanan, M.: On the integrability aspects of the one-dimensional classical continuum isotropic biquadratic Heisenberg spin chain. J. Math. Phys. 33, 1807–1816 (1992)

    Article  ADS  MathSciNet  Google Scholar 

  10. Ankiewicz, A., Clarkson, P.A., Akhmediev, N.: Rogue waves, rational solutions, the patterns of their zeros and integral relations. J. Phys. A: Math. Theor. 43, 122002 (2010). https://doi.org/10.1088/1751-8113/43/12/122002

    Article  ADS  MathSciNet  Google Scholar 

  11. Ankiewicz, A., Akhmediev, N.: Rogue wave solutions for the infinite integrable nonlinear Schrödinger equation hierarchy. Phys. Rev. E 96, 012219 (2017)

    Article  ADS  MathSciNet  CAS  PubMed  Google Scholar 

  12. Ankiewicz, A.: Minimal representation of rogue waves. Wave Motion 112, 102916 (2022)

    Article  MathSciNet  Google Scholar 

  13. Ankiewicz, A.: Rogue and semi-rogue waves defined by volume. Nonlinear Dyn. 104, 4241–4252 (2021). https://doi.org/10.1007/s11071-021-06449-5

    Article  Google Scholar 

  14. Ankiewicz, A., Chowdury, A.: Analysis of characteristics of rogue waves for higher-order equations. Nonlinear Dyn. 109, 1069–1080 (2022). https://doi.org/10.1007/s11071-022-07497-1

    Article  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  16. Ankiewicz, A., Soto-Crespo, J.M., Akhmediev, N.: Rogue waves and rational solutions of the Hirota equation. Phys. Rev. E 81, 046602 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  17. Ankiewicz, A., Bandelow, U., Akhmediev, N.: Generalised Sasa–Satsuma equation: densities approach to new infinite hierarchy of integrable evolution equations. Zeitschrift für Naturforschung A 73(12), 1121–8 (2018). https://doi.org/10.1515/zna-2018-0377

    Article  ADS  CAS  Google Scholar 

  18. Bandelow, U., Ankiewicz, A., Amiranashvili, S., Akhmediev, N.: Sasa–Satsuma hierarchy of integrable evolution equations. Chaos 28, 053108 (2018). https://doi.org/10.1063/1.5030604

    Article  ADS  MathSciNet  CAS  PubMed  Google Scholar 

  19. Kakei, S., Sasa, N., Satsuma, J.: Bilinearization of a generalized derivative nonlinear Schrodinger equation. J. Phys. Soc. Jpn 64, 1519–23 (1995)

    Article  ADS  MathSciNet  CAS  Google Scholar 

  20. Feng, B.-F., Shi, C., Zhang, G., Wu, C.: Higher-order rogue wave solutions of the Sasa–Satsuma equation. J. Phys. A: Math. Theor. 55, 235701 (2022). https://doi.org/10.1088/1751-8121/ac6917

    Article  ADS  MathSciNet  Google Scholar 

  21. Ankiewicz, A., Kedziora, D.J., Akhmediev, N.: Rogue wave triplets. Phys. Lett. A 375, 2782–2785 (2011)

    Article  ADS  CAS  Google Scholar 

Download references

Funding

The author declares that no funds, grants, or other support were received during the preparation of this manuscript.

Author information

Authors and Affiliations

  1. Department of Fundamental & Theoretical Physics, Research School of Physics, The Australian National University, Canberra, ACT, 2601, Australia

    A. Ankiewicz

Authors
  1. A. Ankiewicz
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Single author paper.

Corresponding author

Correspondence to A. Ankiewicz.

Ethics declarations

Conflict of interest

There is 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

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ankiewicz, A. Volumes of second-order rogue waves of the infinite NLS hierarchy. Nonlinear Dyn 112, 3695–3706 (2024). https://doi.org/10.1007/s11071-023-09114-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-023-09114-1

Keywords

Search

Navigation