In optical fibres, weak modulations can grow at the expense of a strong pump to form a triangular comb of sideband pairs, until the process is reversed. Repeated cycles of such conversion and back-conversion constitute a manifestation of the universal nonlinear phenomenon known as Fermi–Pasta–Ulam recurrence. However, it remains a major challenge to observe the coexistence of different types of recurrences owing to the spontaneous symmetry-breaking nature of such a phenomenon. Here, we implement a novel non-destructive technique that allows the evolution in amplitude and phase of frequency modes to be reconstructed via post-processing of the fibre backscattered light. We clearly observe how control of the input modulation seed results in different recursive behaviours emerging from the phase-space structure dictated by the spontaneously broken symmetry. The proposed technique is an important tool to characterize other mixing processes and new regimes of rogue-wave formation and wave turbulence in fibre optics.

Access optionsAccess options

Rent or Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Additional information

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.


  1. 1.

    Fermi, E., Pasta, J. & Ulam, S. in Collected Papers of Enrico Fermi Vol. 2 (ed. Segré, E.) 977–988 (Univ. Chicago Press, Chicago, Illinois, 1965).

  2. 2.

    Porter, M. A., Zabusky, N. J., Hu, B. & Campbell, D. K. Fermi, Pasta, Ulam and the birth of experimental mathematics. Am. Sci. 97, 214–221 (2009).

  3. 3.

    Onorato, M., Vozella, L., Proment, D. & Lvov, Y. V. Route to thermalization in the α-Fermi–Pasta–Ulam system. Proc. Natl Acad. Sci. USA 112, 4208–4213 (2015).

  4. 4.

    Tai, K., Hasegawa, A. & Tomita, A. Observation of modulation instability in optical fibers. Phys. Rev. Lett. 56, 135–138 (1986).

  5. 5.

    Zakharov, V. E. & Ostrovsky, L. A. Modulation instability: the beginning. Phys. D 238, 540–548 (2009).

  6. 6.

    Akhmediev, N. N. Nonlinear physics: déjà vu in optics. Nature 413, 267–268 (2001).

  7. 7.

    Van Simaeys, G., Emplit, Ph. & Haelterman, M. Experimental demonstration of the Fermi–Pasta–Ulam recurrence in a modulationally unstable optical wave. Phys. Rev. Lett. 87, 033902 (2001).

  8. 8.

    Van Simaeys, G., Emplit, Ph. & Haelterman, M. Experimental study of the reversible behaviour of modulational instability in optical fibres. J. Opt. Soc. Am. B 19, 477–486 (2002).

  9. 9.

    Beeckman, J., Hutsebaut, X., Haelterman, M. & Neyts, K. Induced modulation instability and recurrence in nematic liquid crystals. Opt. Express 18, 11185–11195 (2007).

  10. 10.

    Wabnitz, S. & Wetzel, B. Instability and noise-induced thermalization of Fermi–Pasta–Ulam recurrence in the nonlinear Schrödinger equation. Phys. Lett. A 378, 2750–2756 (2014).

  11. 11.

    Guasoni, M. et al. Incoherent Fermi–Pasta–Ulam recurrences and unconstrained thermalization mediated by strong phase correlations. Phys. Rev. X 7, 011025 (2017).

  12. 12.

    Grinevich, P. G. & Santini, P. M. The exact rogue wave recurrence in the NLS periodic setting via matched asymptotic expansions, for 1 and 2 unstable modes. Preprint at https://arxiv.org/abs/1708.04535 (2017).

  13. 13.

    Bao, C. et al. Observation of Fermi–Pasta–Ulam recurrence induced by breather solitons in an optical microresonator. Phys. Rev. Lett. 117, 163901 (2016).

  14. 14.

    Närhi, M. et al. Real-time measurements of spontaneous breathers and rogue wave events in optical fibre modulation instability. Nat. Commun. 7, 13675 (2016).

  15. 15.

    Soto-Crespo, J. M., Devine, N. & Akhmediev, N. Integrable turbulence and rogue waves: breathers or solitons? Phys. Rev. Lett. 116, 103901 (2016).

  16. 16.

    Kibler, B., Chabchoub, A., Gelash, A. N., Akhmediev, N. & Zakharov, V. E. Super-regular breathers in optics and hydrodynamics: omnipresent modulation instability beyond simple periodicity. Phys. Rev. X 5, 041026 (2015).

  17. 17.

    Toenger, S. et al. Emergent rogue wave structures and statistics in spontaneous modulation instability. Sci. Rep. 5, 10380 (2015).

  18. 18.

    Bendhamane, A. et al. Optimal frequency conversion in the nonlinear stage of modulation instability. Opt. Express 23, 30861–30871 (2015).

  19. 19.

    Biondini, G. & Mantzavinos, D. Universal nature of the nonlinear stage of modulational instability. Phys. Rev. Lett. 116, 043902 (2016).

  20. 20.

    Chin, S. A., Ashour, O. A. & Belić, M. R. Anatomy of the Akhmediev breather: cascading instability, first formation time, and Fermi–Pasta–Ulam recurrence. Phys. Rev. E 92, 063202 (2015).

  21. 21.

    Mussot, A., Kudlinski, A., Droques, M., Szriftgiser, P. & Akhmediev, N. Fermi–Pasta–Ulam recurrence in nonlinear fiber optics: the role of reversible and irreversible losses. Phys. Rev. X 4, 011054 (2014).

  22. 22.

    Erkintalo, M. et al. Higher-order modulation instability in nonlinear fibre optics. Phys. Rev. Lett. 107, 253901 (2011).

  23. 23.

    Kibler, B. et al. The Peregrine soliton in nonlinear fibre optics. Nat. Phys. 6, 790–795 (2010).

  24. 24.

    Dudley, J. M., Genty, G., Dias, F., Kibler, B. & Akhmediev, N. Modulation instability, Akhmediev breathers and continuous wave supercontinuum generation. Opt. Express 17, 21497–21508 (2009).

  25. 25.

    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).

  26. 26.

    Dudley, J. M., Dias, F., Erkintalo, M. & Genty, G. Instabilities, breathers and rogue waves in optics. Nat. Photon. 8, 755–764 (2014).

  27. 27.

    Akhmediev, N. N., Eleonoskii, V. M. & Kulagin, N. E. Generation of periodic trains of picosecond pulses in an optical fibre: exact solutions. Sov. Phys. JETP 62, 894–899 (1985).

  28. 28.

    Akhmediev, N. N., Eleonskii, V. M. & Kulagin, N. E. Exact first-order solutions of the nonlinear Schrödinger equation. Theor. Math. Phys. 72, 809–818 (1987).

  29. 29.

    Ablowitz, M. J. & Herbst, B. M. On homoclinic structure and numerically induced chaos for the nonlinear Schrödinger equation. SIAM J. Appl. Math. 50, 339–351 (1990).

  30. 30.

    Moon, H. T. Homoclinic crossings and pattern selection. Phys. Rev. Lett. 64, 412–414 (1990).

  31. 31.

    Trillo, S. & Wabnitz, S. Dynamics of the modulational instability in optical fibers. Opt. Lett. 16, 986–988 (1991).

  32. 32.

    Trillo, S. & Wabnitz, S. Self-injected spatial mode locking and coherent all-optical FM/AM switching based on modulational instability. Opt. Lett. 16, 1566–1568 (1991).

  33. 33.

    Liu, C. Spontaneous symmetry breaking and chance in a classical world. Philos. Sci. 70, 590–608 (2003).

  34. 34.

    Kimmoun, O. et al. Modulation Instability and phase-shifted Fermi–Pasta–Ulam recurrence. Sci. Rep. 6, 28516 (2016).

  35. 35.

    Ablowitz, M. J., Hammack, J., Henderson, D. & Schober, C. M. Modulated periodic Stokes waves in deep water. Phys. Rev. Lett. 84, 887–890 (2000).

  36. 36.

    Cappellini, G. & Trillo, S. Third-order three-wave mixing in single-mode fibres: exact solutions and spatial instability effects. J. Opt. Soc. Am. B 8, 824–834 (1991).

  37. 37.

    Butikov, E. I. The rigid pendulum—an antique but evergreen physical model. Eur. J. Phys. 20, 429–441 (1999).

  38. 38.

    Healey, P. Fading in heterodyne OTDR. Electron. Lett. 20, 30–32 (1984).

  39. 39.

    Deng, G., Li, S., Biondini, G. & Trillo, S. Recurrence due to periodic multisoliton fission in the defocusing nonlinear Schrödinger equation. Phys. Rev. E 96, 052213 (2017).

Download references


This work was partly supported by the Agence Nationale de la Recherche through the High Energy All Fiber Systems (HEAFISY) and Nonlinear dynamics of Abnormal Wave Events (NoAWE) projects, the Labex Centre Europeen pour les Mathematiques, la Physique et leurs Interactions (CEMPI) and Equipex Fibres optiques pour les hauts flux (FLUX) through the ‘Programme Investissements d’Avenir’, by the Ministry of Higher Education and Research, Hauts de France council and European Regional Development Fund (ERDF) through the Contrat de Projets Etat-Region (CPER Photonics for Society, P4S) and FEDER through the HEAFISY project.

Author information


  1. University of Lille, CNRS, UMR 8523 - PhLAM - Physique des Lasers Atomes et Molécules, Lille, France

    • Arnaud Mussot
    • , Corentin Naveau
    • , Matteo Conforti
    • , Alexandre Kudlinski
    • , Francois Copie
    •  & Pascal Szriftgiser
  2. Department of Engineering, University of Ferrara, Ferrara, Italy

    • Stefano Trillo


  1. Search for Arnaud Mussot in:

  2. Search for Corentin Naveau in:

  3. Search for Matteo Conforti in:

  4. Search for Alexandre Kudlinski in:

  5. Search for Francois Copie in:

  6. Search for Pascal Szriftgiser in:

  7. Search for Stefano Trillo in:


A.M. and P.S. conceived the experimental setup. A.M., C.N., A.K., F.C. and P.S. worked on the experiment. M.C. and S.T. developed the theoretical aspects. A.M., S.T., M.C. and C.N. performed numerical simulations. All authors contributed to analysing the data and writing the paper.

Competing interests

The authors declare no competing interests.

Corresponding authors

Correspondence to Arnaud Mussot or Stefano Trillo.

Supplementary information

  1. Supplementary Information

    Supplementary notes and figures, and references.

About this article

Publication history




Issue Date



Further reading