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Multi-elliptic rogue wave clusters of the nonlinear Schrödinger equation on different backgrounds

Nonlinear Dynamics volume 108, pages 479–490 (2022)Cite this article

Abstract

In this work, we analyze the multi-elliptic rogue wave clusters of the nonlinear Schrödinger equation (NLSE) in order to understand more thoroughly the origin and appearance of optical rogue waves in this system. Such structures are obtained on uniform backgrounds by using the Darboux transformation scheme for finding analytical solutions of the NLSE under various conditions. In particular, we solve the eigenvalue problem of the Lax pair of order n in which the first m evolution shifts are equal, nonzero, and eigenvalue dependent, while the imaginary parts of all eigenvalues tend to one. We show that an Akhmediev breather of order \(n-2m\) appears at the origin of the (x, t) plane and can be considered as the central rogue wave of the so-formed cluster. We show that the high-intensity narrow peak, with the characteristic intensity distribution in its vicinity, is enclosed by m ellipses consisting of the first-order Akhmediev breathers. The number of maxima on each ellipse is determined by its index and the solution order. Since rogue waves in nature usually appear on a wavy background, we utilize the modified Darboux transformation scheme to build such solutions on a Jacobi elliptic dnoidal background. We analyze the vertical semi-axis of all ellipses in a cluster as a function of an absolute evolution shift. We show that the cluster radial symmetry in the (x, t) plane is broken when the shift value is increased above a threshold. We apply the same analysis on the Hirota equation, to examine the influence of a real parameter and Hirota’s operator on the cluster appearance. The same analysis can be applied to the infinite hierarchy of extended NLSEs. The main outcomes of this paper are the new multi-rogue wave solutions of the nonlinear Schrödinger equation and its extended family on uniform and elliptic backgrounds.

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All data generated or analyzed during this study are included in the published article.

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Acknowledgements

This research is supported by the Qatar National Research Fund (a member of Qatar Foundation). S.N.N. acknowledges funding provided by the Institute of Physics Belgrade, through the grant by the Ministry of Education, Science, and Technological Development of the Republic of Serbia. S.A. is supported by the Embassy of Libya in the Republic of Serbia. O.A.A. is supported by the Berkeley Graduate Fellowship and the Anselmo J. Macchi Graduate Fellowship. N.B.A. acknowledges support from Project No. 18-11-00247 of the Russian Science Foundation. M.R.B. acknowledges support by the Al-Sraiya Holding Group.

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Authors and Affiliations

  1. Science Program, Texas A&M University at Qatar, P.O. Box 23874, Doha, Qatar

    Stanko N. Nikolić

  2. Institute of Physics Belgrade, University of Belgrade, Pregrevica 118, Belgrade, 11080, Serbia

    Stanko N. Nikolić

  3. Faculty of Physics, University of Belgrade, Studentski trg 12, Belgrade, 11001, Serbia

    Sarah Alwashahi

  4. Faculty of Science Al-Zawiya, University of Libya, Al Ajaylat, Libya

    Sarah Alwashahi

  5. Department of Physics, University of California, Berkeley, CA, 94720, USA

    Omar A. Ashour

  6. Department of Physics and Astronomy, Texas A&M University, College Station, TX, 77843, USA

    Siu A. Chin

  7. Moscow State Technological University “STANKIN”, Moscow, Russia

    Najdan B. Aleksić

  8. Science Program, Texas A&M University at Qatar, P.O. Box 23874, Doha, Qatar

    Najdan B. Aleksić & Milivoj R. Belić

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  1. Stanko N. Nikolić
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  2. Sarah Alwashahi
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  3. Omar A. Ashour
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Nikolić, S.N., Alwashahi, S., Ashour, O.A. et al. Multi-elliptic rogue wave clusters of the nonlinear Schrödinger equation on different backgrounds. Nonlinear Dyn 108, 479–490 (2022). https://doi.org/10.1007/s11071-021-07194-5

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Keywords

  • Nonlinear Schrödinger equation
  • Rogue waves
  • Circular and triangular rogue wave clusters
  • Darboux transformation