Skip to main content
SpringerLink
Log in

Rogue waves formation by solitons synchronization and resonance: Gerdjikov-Ivanov equation

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

Abstract

Rogue waves usually possess localized large-amplitude waves and coming from nowhere and disappearing without any trace. The Peregrine solitons with the above characteristics are regarded as rogue waves prototype. The paper mainly studies the effects of multiple solitons in the formation of rogue waves for the Gerdjikov-Ivanov equation on vanishing boundary and nonvanishing boundary conditions by the N-fold Darboux transformation with the pure imaginary spectral parameters: (i) under the vanishing boundary conditions, all spectral parameters close to one value, and combined with the number of Darboux transformations, the solitons synchronization interactions can generate Peregrine-like solitons. (ii) under the nonvanishing boundary conditions, the spectral parameters \(\lambda _k \rightarrow \lambda _{c_1}(\lambda _{c_2})\), where \(\lambda _{c_1}\) and \(\lambda _{c_2}\) are a pair of key eigenvalues associated with the phase resonance of the solitons, the solitons resonance and degradation interactions enable to obtain Peregrine-like solitons, which can be considered as the generalization of Peregrine solitons. These results also well illustrate the relationship between the generation of rogue waves and spectral parameters and boundary conditions and provide a new idea for rogue waves excitation.

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

Similar content being viewed by others

Data availability

The data that support the findings of the study are available within the article.

References

  1. Onorato, M., Residori, S., Bortolozzo, U., Montina, A., Arecchi, F.T.: Rogue waves and their generating mechanisms in different physical contexts. Phys. Rep. 528, 47–89 (2013)

    MathSciNet  Google Scholar 

  2. Kharif, C., Pelinovsky, E., Slunyaev, A.: Rogue waves in the ocean (Berlin: Springer). (2009)

  3. Chabchoub, A., Hoffmann, N.P., Akhmediev, N.: Rogue Wave Observation in a Water Wave Tank. Phys. Rev. Lett. 106, 204502 (2011)

    Google Scholar 

  4. Chabchoub, A., Hoffmann, N., Onorato, M., Akhmediev, N.: Super Rogue Waves: Observation of a Higher-Order Breather in Water Waves. Phys. Rev. X 2, 011015 (2012)

    Google Scholar 

  5. Solli, D.R., Ropers, C., Koonath, P., Jalali, B.: Optical rogue waves. Nature 450, 1054–1057 (2007)

    Google Scholar 

  6. Kibler, B., Fatome, J., Finot, C., Millot, G., Dias, F., Genty, G., Akhmediev, N., Dudley, J.M.: The Peregrine soliton in nonlinear fibre optics. Nat. Phys. 6, 790–795 (2010)

    Google Scholar 

  7. Zakharov, V.E., Ostrovsky, L.A.: Modulation instability: the beginning. Physica D 238, 540 (2009)

    MathSciNet  MATH  Google Scholar 

  8. Zakharov, V.E., Gelash, A.A.: Nonlinear stage of modulation instability. Phys. Rev. Lett. 111, 054101 (2013)

    Google Scholar 

  9. Peregrine, D.H.: Water waves, Nonlinear Schrödinger equations and their solutions. J. Aust. Math. Soc. Ser. B, Appl. Math. 25, 16–43 (1983)

    MATH  Google Scholar 

  10. Akhmediev, N., Ankiewicz, A., Taki, M.: Waves that appear from nowhere and disappear without a trace. Phys. Lett. A 373, 675–678 (2009)

    MATH  Google Scholar 

  11. Kuznetsov, E.A.: Solitons in a parametrically unstable plasma. Sov. Phys.-Dokl. 22, 507–508 (1977)

    Google Scholar 

  12. Ma, Y.C.: The perturbed plane-wave solutions of the cubic Schröinger equation. Stud. Appl. Math. 60, 43–58 (1979)

    MathSciNet  Google Scholar 

  13. Akhmediev, N.N., Korneev, V.I.: Modulation instability and periodic solutions of nonlinear Schrödinger equation. Theor. Math. Phys. 69, 1089–1093 (1987)

    MATH  Google Scholar 

  14. Xu, S.W., He, J.S., Wang, L.H.: The Darboux transformation of the derivative nonlinear Schrödinger equation. J. Phys. A: Math. Theor. 44, 305203 (2011)

    MATH  Google Scholar 

  15. Xu, S.W., He, J.S.: The rogue wave and breather solution of the Gerdjikov-Ivanov equation. J. Math. Phys. 53, 063507 (2012)

    MathSciNet  MATH  Google Scholar 

  16. Guo, B.L., Ling, L.M., Liu, Q.P.: High-order solutions and generalized darboux transformations of derivative nonlinear Schrödinger equations. Stud. Appl. Math. 130, 317–344 (2012)

    MATH  Google Scholar 

  17. He, J.S., Wang, L.H., Li, L.J., Porsezian, K., Erdélyi, R.: Few-cycle optical rogue waves: complex modified Korteweg-de Vries equation. Phys. Rev. E 89, 062917 (2014)

    Google Scholar 

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

    MathSciNet  Google Scholar 

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

    Google Scholar 

  20. Ankiewicz, A.: Rogue and semi-rogue waves defined by volume. Nonlinear Dyn. 104, 4241–4252 (2021)

    Google Scholar 

  21. Bertola, M., El, G.A., Tovbis, A.: Rogue waves in multiphase solutions of the focusing nonlinear Schrödinger equation. Proc. R. Soc. A 472, 20160340 (2016)

    MATH  Google Scholar 

  22. Xu, S.W., He, J.S., Mihalache, D.: Rogue waves generation through multiphase solutions degeneration for the derivative nonlinear Schrodinger equation. Nonlinear Dyn. 97, 2443–2452 (2019)

    MATH  Google Scholar 

  23. Chen, J.B., Pelinovsky, D.E.: Rogue waves on the background of periodic standing waves in the derivative nonlinear Schrodinger equation. Phys. Rev. E 103, 062206 (2021)

    MathSciNet  Google Scholar 

  24. Yang, B., Yang, J.K.: Partial-rogue waves that come from nowhere but leave with a trace in the Sasa-Satsuma equation. Phys. Lett. A 458, 128573 (2023)

    MathSciNet  MATH  Google Scholar 

  25. Slunyaev, A.V., Pelinovsky, E.N.: Role of multiple soliton interactions in the generation of rogue waves: the modified Korteweg-de vries framework. Phys. Rev. Lett. 117, 214501 (2016)

    Google Scholar 

  26. Xu, S.W., Wang, L.H., Erdélyi, R., He, J.S.: Degeneracy in bright-dark solitons of the derivative nonlinear Schrodinger equation. Appl. Math. Lett. 87, 64–72 (2019)

    MathSciNet  MATH  Google Scholar 

  27. Guo, L.J., Chabchoub, A., He, J.S.: Higher-order rogue wave solutions to the Kadomtsev-Petviashvili I equation. Physica D 426, 132990 (2021)

    MathSciNet  MATH  Google Scholar 

  28. Rao, J.G., He, J.S., Malomed, B.A.: Resonant collisions between lumps and periodic solitons in the Kadomtsev-Petviashvili I equation. J. Math. Phys. 63, 013510 (2022)

    MathSciNet  MATH  Google Scholar 

  29. Pelinovsky, E.N., Shurgalina, E.G., Sergeeva, A.V., Talipova, T.G., El, G.A., Grimshaw, R.H.J.: Two-soliton interaction as an elementary act of soliton turbulence in integrable systems. Phys. Lett. A 377, 272–275 (2013)

    MATH  Google Scholar 

  30. Herink, G., Kurtz, F., Jalali, B., Solli, D.R., Ropers, C.: Real-time spectral interferometry probes the internal dynamics of femtosecond soliton molecules. Science 356, 50–54 (2017)

    Google Scholar 

  31. Liu, X., Yao, X., Cui, Y.: Real-time observation of the buildup of soliton molecules. Phys. Rev. Lett. 121, 023905 (2018)

    Google Scholar 

  32. Melchert, O., Willms, S., Bose, S., Yulin, A., Roth, B., Mitschke, F., Morgner, U., Babushkin, I., Demircan, A.: Soliton molecules with two frequencies. Phys. Rev. Lett. 123, 243905 (2019)

    Google Scholar 

  33. Xu, G., Gelash, A., Chabchoub, A., Zakharov, V., Kibler, B.: Breather wave molecules. Phys. Rev. Lett. 122, 084101 (2019)

    Google Scholar 

  34. Lou, S.Y.: Soliton molecules and asymmetric solitons in three fifth order systems via velocity resonance. J. Phys. Commun. 4, 041002 (2020)

    Google Scholar 

  35. Jaradat, I., Alquran, M., Qureshi, S., Sulaiman, T.A., Yusuf, A.: Convex-rogue, half-kink, cusp-soliton and other bidirectional wave-solutions to the generalized Pochhammer-Chree equation. Phys. Scr. 97, 055203 (2022)

    Google Scholar 

  36. Yusuf, A., Alshomrani, A.S., Sulaiman, T.A., Isah, I., Baleanu, D.: Extended classical optical solitons to a nonlinear Schrödinger equation expressing the resonant nonlinear light propagation through isolated flaws in optical waveguides. Opt. Quant. Electron. 54, 853 (2022)

    Google Scholar 

  37. Ibrahim, S., Sulaiman, T.A., Yusuf, A., Alshomrani, A.S., Baleanu, D.: Families of optical soliton solutions for the nonlinear Hirota-Schrödinger equation. Opt. Quant. Electron. 54, 722 (2022)

    Google Scholar 

  38. Yusuf, A., Sulaiman, T.A., Alshomrani, A.S., Baleanu, D.: Optical solitons with nonlinear dispersion in parabolic law medium and three-component coupled nonlinear Schrödinger equation. Opt. Quant. Electron. 54, 390 (2022)

    Google Scholar 

  39. Osman, M.S., Tariq, K.U., Bekir, A., Elmoasry, A., Elazab, N., Younis, M., Abdel-Aty, M.: Investigation of soliton solutions with different wave structures to the (2+1)-dimensional Heisenberg ferromagnetic spin chain equation. Commun. Theor. Phys. 72, 035002 (2020)

    MathSciNet  MATH  Google Scholar 

  40. Malik, S., Almusawa, H., Kumar, S., Wazwaz, A.M., Osman, M.S.: A (2+1)-dimensional Kadomtsev-Petviashvili equation with competing dispersion effect: Painlevé analysis, dynamical behavior and invariant solutions. Results Phys. 23, 104043 (2021)

    Google Scholar 

  41. Yiasir Arafat, S.M., Fatema, K., Rayhanul Islam, S.M., Ekramul Islam, Md., Ali Akbar, M., Osman, M.S.: The mathematical and wave profile analysis of the Maccari system in nonlinear physical phenomena. Opt. Quant. Electron. 55, 136 (2023)

    Google Scholar 

  42. Ismael, H.F., Sulaiman, T.A., Osman, M.S.: Multi-solutions with specific geometrical wave structures to a nonlinear evolution equation in the presence of the linear superposition principle. Commun. Theor. Phys. 75, 015001 (2023)

    MathSciNet  Google Scholar 

  43. Yusuf, A., Sulaiman, T.A., Alshomrani, A.S., Baleanu, D.: Breather and lump-periodic wave solutions to a system of nonlinear wave model arising in fluid mechanics. Nonlinear Dyn. 110, 3655–3669 (2022)

    Google Scholar 

  44. Ismael, H.F., Sulaiman, T.A., Nabi, H.R., Shah, N.A., Botmart, T.: Multiple soliton, M-lump and interaction solutions to the (3+1)-dimensional soliton equation. Results Phys. 45, 106220 (2023)

    Google Scholar 

  45. Hosseini, K., Hincal, E., Baleanu, D., Obi, O.A., Salahshour, S.: Non-singular multi-complexiton wave to a generalized KdV equation. Nonlinear Dyn. (2023). https://doi.org/10.1007/s11071-022-08208-6

    Article  Google Scholar 

  46. Kaup, D.J., Newell, A.C.: An exact solution for a derivative nonlinear Schrödinger equation. J. Math. Phys. 19, 798–801 (1978)

    MATH  Google Scholar 

  47. Chen, H.H., Lee, Y.C., Liu, C.S.: Integrability of nonlinear hamiltonian systems by inverse scattering method. Phys. Scr. 20, 490–492 (1979)

    MathSciNet  MATH  Google Scholar 

  48. Gerdjikov, V.S., Ivanov, I.: A quadratic pencil of general type and nonlinear evolution equations. ii. hierarchies of hamiltonian structures. J. Phys. Bulgar. 10, 130–143 (1983)

    MathSciNet  Google Scholar 

  49. Fan, E.G.: Darboux transformation and soliton-like solutions for the Gerdjikov-Ivanov equation. J. Phys. A 33, 6925–6933 (2000)

    MathSciNet  MATH  Google Scholar 

  50. Fan, E.G.: Integrable evolution systems based on Gerdjikov-Ivanov equation, bi-Hamiltonian structure, finite-dimensional integrable systems and N-fold darboux transformation. J. Math. Phys. 41, 7769–7782 (2000)

    MathSciNet  MATH  Google Scholar 

  51. Yu, J., He, J.S., Han, J.W.: Two kinds of new integrable decompositions of the Gerdjikov-Ivanov equation. J. Math. Phys. 53, 033510 (2012)

    MathSciNet  MATH  Google Scholar 

  52. Xu, J., Fan, E.G., Chen, Y.: Long-time asymptotic for the derivative nonlinear Schrodinger equation with step-like initial value. Math. Phys. Anal. Geom. 16, 253–288 (2013)

    MathSciNet  MATH  Google Scholar 

  53. Tian, S.F., Zhang, T.T.: Long-time asymptotic behavior for the Gerdjikov-Ivanov type of derivative nonlinear Schrodinger equation with time-periodic boundary condition. P. Am. Math. Soc. 146, 1713–1729 (2018)

    MathSciNet  MATH  Google Scholar 

  54. Zhang, Z.C., Fan, E.G.: Inverse scattering transform and multiple high-order pole solutions for the Gerdjikov-Ivanov equation under the zero/nonzero background. Z. Angew. Math. Phys. 72, 153–177 (2021)

  55. Guo, L.J., Zhang, Y.S., Xu, S.W., Wu, Z.W.: The higher order rogue wave solutions of the Gerdjikov-Ivanov equation. Phys. Scr. 89, 035501 (2014)

    Google Scholar 

  56. Nie, H., Zhu, J.Y., Geng, X.G.: Trace formula and new form of N-soliton to the Gerdjikov-Ivanov equation. Anal. Math. Phys. 8, 415–426 (2018)

    MathSciNet  MATH  Google Scholar 

  57. Zhang, S.S., Xu, T., Li, M., Zhang, X.F.: Higher-order algebraic soliton solutions of the Gerdjikov-Ivanov equation: asymptotic analysis and emergence of rogue waves. Physica D 432, 133128 (2022)

    MathSciNet  MATH  Google Scholar 

Download references

Funding

This work is supported by the National Natural Science Foundation of China under Grant No. 11601187 and 12171433.

Author information

Authors and Affiliations

  1. College of Data Science, Jiaxing University, Jiaxing, 314001, Zhejiang, People’s Republic of China

    Zitian Li & Shuwei Xu

  2. School of Science, Zhejiang University of Science and Technology, Hangzhou, 310023, Zhejiang, People’s Republic of China

    Yongshuai Zhang

Authors
  1. Zitian Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shuwei Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yongshuai Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shuwei Xu.

Ethics declarations

Conflict of interest

The authors declare that there are no conflicts of interests regarding the publication of the paper.

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

Li, Z., Xu, S. & Zhang, Y. Rogue waves formation by solitons synchronization and resonance: Gerdjikov-Ivanov equation. Nonlinear Dyn 111, 11447–11458 (2023). https://doi.org/10.1007/s11071-023-08426-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-023-08426-6

Keywords

Search

Navigation