Theoretical and Mathematical Physics

, Volume 196, Issue 3, pp 1294–1306 | Cite as

Phase Resonances of the NLS Rogue Wave Recurrence in the Quasisymmetric Case

  • P. G. GrinevichEmail author
  • P. M. Santini


Based on experimental observations of the recurrence of anomalous waves in water and nonlinear optics, we investigate the theory of anomalous waves for initial data almost satisfying the symmetry conditions in the experiment. We also derive useful formulas, in particular, describing the phase resonance in the recurrence, which can be compared with both the currently available experimental data and the experimental data to be obtained in the near future.


focusing nonlinear Schrödinger equation anomalous wave recurrence almost symmetric configuration phase resonance 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    V. E. Zakharov, “Stability of periodic waves of finite amplitude on the surface of a deep fluid,” J. Appl. Mech. Tech. Phys., 9, 190–194 (1968).ADSCrossRefGoogle Scholar
  2. 2.
    D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature, 450, 1054 (2007).ADSCrossRefGoogle Scholar
  3. 3.
    U. Bortolozzo, A. Montina, F. T. Arecchi, J. P. Huignard, and S. Residori, “Spatiotemporal pulses in a liquid crystal optical oscillator,” Phys. Rev. Lett., 99, 023901 (2007).ADSCrossRefGoogle Scholar
  4. 4.
    D. Pierangeli, F. Di Mei, C. Conti, A. J. Agranat, and E. DelRe, “Spatial rogue waves in photorefractive ferroelectrics,” Phys. Rev. Lett., 115, 093901 (2015).ADSCrossRefGoogle Scholar
  5. 5.
    C. Sulem and P-L. Sulem, The Nonlinear Schrödinger Equation: Self Focusing and Wave Collapse (Appl. Math. Sci., Vol. 139), Springer, New York (1999).zbMATHGoogle Scholar
  6. 6.
    Yu. V. Bludov, V. V. Konotop, and N. Akhmediev, “Matter rogue waves,” Phys. Rev. A, 80, 033610 (2009).ADSCrossRefGoogle Scholar
  7. 7.
    G. G. Stokes, “On the theory of oscillatory waves,” in: Mathematical and Physical Papers (Trans. Cambridge Phil. Soc., Vol. 8), Cambridge Univ. Press, Cambridge (2009), pp. 441–455; “Supplement to a paper on the theory of oscillatory waves,” in: Mathematical and Physical Papers (Trans. Cambridge Phil. Soc., Vol. 1), Cambridge Univ. Press, Cambridge (1880), pp. 314–326.Google Scholar
  8. 8.
    V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett., 3, 307–310 (1966).ADSGoogle Scholar
  9. 9.
    T. B. Benjamin and J. E. Feir, “The disintegration of wave trains on deep water: Part I. Theory,” J. Fluid Mech., 27, 417–430 (1967).ADSCrossRefzbMATHGoogle Scholar
  10. 10.
    V. Zakharov and L. Ostrovsky, “Modulation instability: The beginning,” Phys. D, 238, 540–548 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    T. Taniuti and H. Washimi, “Self-trapping and instability of hydromagnetic waves along the magnetic field in a cold plasma,” Phys. Rev. Lett., 21, 209–212 (1968).ADSCrossRefGoogle Scholar
  12. 12.
    L. Salasnich, A. Parola, and L. Reatto, “Modulational instability and complex dynamics of confined matter-wave solitons,” Phys. Rev. Lett., 91, 080405 (2003); arXiv:cond-mat/030720v1 (2003).ADSCrossRefGoogle Scholar
  13. 13.
    K. L. Henderson, D. H. Peregrine, and J. W. Dold, “Unsteady water wave modulations: Fully nonlinear solutions and comparison with the nonlinear Schrödinger equation,” Wave Motion, 29, 341–361 (1999).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    K. B. Dysthe and K. Trulsen, “Note on breather type solutions of the NLS as models for freak-waves,” Phys. Scr., T 82, 48–52 (1999).ADSCrossRefGoogle Scholar
  15. 15.
    A. Osborne, M. Onorato, and M. Serio, “The nonlinear dynamics of rogue waves and holes in deep-water gravity wave trains,” Phys. Lett. A, 275, 386–393 (2000).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    C. Kharif and E. Pelinovsky, “Physical mechanisms of the rogue wave phenomenon,” Eur. J. Mech. B Fluids, 22, 603–634 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    C. Kharif, E. Pelinovsky, T. Talipova, and A. Slunyaev, “Focusing of nonlinear wave groups in deep water,” JETP Lett., 73, 170–175 (2001).ADSCrossRefzbMATHGoogle Scholar
  18. 18.
    M. Onorato, S. Residori, U. Bortolozzo, A. Montina, and F. T. Arecchi, “Rogue waves and their generating mechanisms in different physical contexts,” Phys. Rep., 528, 47–89 (2013).ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    V. F. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional selfmodulation of wave in nonlinear media,” JETP, 34, 62–69 (1972).ADSMathSciNetGoogle Scholar
  20. 20.
    A. R. Its, A. V. Rybin, and M. A. Sall’, “Exact integration of nonlinear Schrödinger equation,” Theor. Math. Phys., 74, 20–32 (1988).CrossRefzbMATHGoogle Scholar
  21. 21.
    E. D. Belokolos, A. I. Bobenko, V. Z. Enol’ski, A. R. Its, and V. B. Matveev, Algebro-Geometric Approach in the Theory of Integrable Equations, Springer, Berlin (1994).Google Scholar
  22. 22.
    I. M. Krichever, “Spectral theory of two-dimensional periodic operators and its applications,” Russ. Math. Surveys, 44, 145–225 (1989).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    I. M. Krichever, “Perturbation theory in periodic problems for two-dimensional integrable systems,” Sov. Sci. Rev. Sect. C Math. Phys. Rev., 9, 1–103 (1992).zbMATHGoogle Scholar
  24. 24.
    V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons, Springer, Berlin (1991).CrossRefzbMATHGoogle Scholar
  25. 25.
    N. Ercolani, M. G. Forest, and D. W. McLaughlin, “Geometry of the modulation instability Part III: Homoclinic orbits for the periodic sine-Gordon equation,” Phys. D, 43, 349–384 (1990).MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    V. E. Zakharov and A. B. Shabat, “A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem: I,” Funct. Anal. Appl., 8, 226–235 (1974).CrossRefzbMATHGoogle Scholar
  27. 27.
    V. E. Zakharov and A. V. Mikhailov, “Relativistically invariant two-dimensional models of field theory which are integrable by means of the inverse scattering problem method,” JETP, 47, 1017–1027.Google Scholar
  28. 28.
    D. H. Peregrine, “Water waves, nonlinear Schrödinger equations, and their solutions,” J. Austral. Math. Soc. Ser. B, 25, 16–43 (1983).MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    E. A. Kuznetsov, “Solitons in a parametrically unstable plasma,” Sov. Phys. Dokl., 22, 507–508 (1977).ADSGoogle Scholar
  30. 30.
    T. Kawata and H. Inoue, “Inverse scattering method for the nonlinear evolution equations under nonvanishing conditions,” J. Phys. Soc. Japan, 44, 1722–1729 (1978).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Y.-C. Ma, “The perturbed plane wave solutions of the cubic Schrödinger equation,” Stud. Appl. Math., 60, 43–58 (1979).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    N. N. Akhmedieva, V. M. Eleonskii, and N. E. Kulagin, “Generation of periodic trains of picoseconld pulses in an optical fiber: Exact solutions,” JETP, 62, 894–899 (1985).Google Scholar
  33. 33.
    N. N. Akhmediev and V. I. Korneev, “Modulation instability and periodic solutions of the nonlinear Schrödinger equation,” Theor. Math. Phys., 69, 1089–1093 (1986).CrossRefzbMATHGoogle Scholar
  34. 34.
    N. N. Akhmediev, V. M. Eleonskii, and N. E. Kulagin, “Exact first-order solutions of the nonlinear Schrödinger equation,” Theor. Math. Phys., 72, 809–818 (1987).CrossRefzbMATHGoogle Scholar
  35. 35.
    V. E. Zakharov and A. A. Gelash, “Soliton on unstable condensate,” arXiv:1109.06201109.0620 (2011).Google Scholar
  36. 36.
    P. Dubard, P. Gaillard, C. Klein, and V. B. Matveev, “On multi-rogue wave solutions of the NLS equation and positon solutions of the KdV equation,” Eur. Phys. J. Special Topics, 185, 247–258 (2010).ADSCrossRefGoogle Scholar
  37. 37.
    R. Hirota, Direct Methods for Finding Exact Solutions of Nonlinear Evolution Equations (Lect. Notes Math., Vol. 515), Springer, New York (1976).CrossRefzbMATHGoogle Scholar
  38. 38.
    D. J. Kedziora, A. Ankiewicz, and N. Akhmediev, “Second-order nonlinear Schrödinger equation breather solutions in the degenerate and rogue wave limits,” Phys. Rev. E, 85, 066601 (2012).ADSCrossRefGoogle Scholar
  39. 39.
    V. E. Zakharov and A. A. Gelash, “On the nonlinear stage of modulation instability,” Phys. Rev. Lett., 111, 054101 (2013).ADSCrossRefGoogle Scholar
  40. 40.
    F. Baronio, A. Degasperis, M. Conforti, and S. Wabnitz, “Solutions of the vector nonlinear Schrödinger equations: Evidence for deterministic rogue waves,” Phys. Rev. Lett., 109, 044102 (2012).ADSCrossRefGoogle Scholar
  41. 41.
    A. Degasperis and S. Lombardo, “Integrability in action: Solitons, instability, and rogue waves,” in: Rogue and Shock Waves in Nonlinear Dispersive Media (Lect. Notes Phys.,Vol. 926, M. Onorato, S. Resitori, and F. Baronio, eds.), Springer, Cham (2016).Google Scholar
  42. 42.
    A. Degasperis, S. Lombardo, and M. Sommacal, “Integrability and linear stability of nonlinear waves,” J. Nonlinear Sci., 28, 1251–1291 (2018); arXiv:1707.09536v2 [nlin.SI] (2017).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    G. Biondini and G. Kovacic, “Inverse scattering transform for the focusing nonlinear Schrödinger equation with nonzero boundary conditions,” J. Math. Phys., 55, 031506 (2014).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    G. Biondini, S. Li, and D. Mantzavinos, “Oscillation structure of localized perturbations in modulationally unstable media,” Phys. Rev. E, 94, 060201 (2016).ADSCrossRefGoogle Scholar
  45. 45.
    S. P. Novikov, “The periodic problem for the Korteweg–de Vries equation,” Funct. Anal. Appl., 8, 236–246 (1974).MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    A. R. Its and V. P. Kotljarov, “Explicit formulas for solutions of a nonlinear Schrödinger equation [in Russian],” Dokl. Akad. Nauk Ukrain. SSR Ser. A, 11, 965–968, 1051 (1976).Google Scholar
  47. 47.
    I. M. Krichever, “Methods of algebraic geometry in the theory of non-linear equations,” Russ. Math. Surveys, 32, 185–213 (1977).ADSCrossRefzbMATHGoogle Scholar
  48. 48.
    P. G. Grinevich and P. M. Santini, “The finite gap method and the analytic description of the exact rogue wave recurrence in the periodic NLS Cauchy problem: 1,” arXiv:1707.05659v2 [nlin.SI] (2017).Google Scholar
  49. 49.
    P. G. Grinevich and P. M. Santini, “The finite gap method and the solution of the rogue wave periodic Cauchy problem in the case of a finite number of unstable modes,” Preprint (2018 in preparation).Google Scholar
  50. 50.
    P. G. Grinevich and P. M. Santini, “The exact rogue wave recurrence in the NLS periodic setting via matched asymptotic expansions, for 1 and 2 unstable modes,” Phys. Lett. A, 382, 973–979 (2018); arXiv:1708.04535v1 [nlin.SI] (2017).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    E. R. Tracy and H. H. Chen, “Nonlinear self-modulation: An exactly solvable model,” Phys. Rev. A, 37, 815–839 (1988).ADSMathSciNetCrossRefGoogle Scholar
  52. 52.
    J. Javanainen and J. Ruostekoski, “Symbolic calculation in development of algorithms: Split-step methods for the Gross–Pitaevskii equation,” J. Phys. A: Math. Gen., 39, L179–L184 (2006); arXiv:cond-math/0411154v1 (2004).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    G. P. Agrawal, Nonlinear Fiber Optics, Acad. Press, New York (2001).zbMATHGoogle Scholar
  54. 54.
    J. A. C. Weideman and B. M. Herbst, “Split-step methods for the solution of the nonlinear Schrödinger equation,” SIAM J. Numer. Anal., 23, 485–507 (1986).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    T. R. Taha and X. Xu, “Parallel split-step fourier methods for the coupled nonlinear Schrödinger type equations,” J. Supercomp., 32, 5–23 (2005).CrossRefGoogle Scholar
  56. 56.
    P. G. Grinevich and P. M. Santini, “Numerical instability of the Akhmediev breather and a finite-gap model of it,” arXiv:1708.00762v2 [nlin.PS] (2017); V. M. Buchstaber, S. Konstantinou-Rizo, and A. V. Mikhailov, eds., Recent Developments in Integrable Systems and Related Topics of Mathematical Physics, Springer, New York (2018).Google Scholar
  57. 57.
    M. J. Ablowitz and Z. H. Musslimani, “Integrable nonlocal nonlinear Schrödinger equation,” Phys. Rev. Lett., 110, 064105 (2013).ADSCrossRefGoogle Scholar
  58. 58.
    M. J. Ablowitz and Z. H. Musslimani, “Integrable discrete PT symmetric model,” Phys. Rev. Lett. E, 90, 032912 (2014).ADSCrossRefGoogle Scholar
  59. 59.
    M. J. Ablowitz and Z. H. Musslimani, “Integrable nonlocal nonlinear equations,” Stud. Appl. Math., 139, 7–59 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    P. M. Santini, “Darboux-dressing and symmetry construction of classes of regular and singular solutions of the NLS and the PT-symmetric NLS equations over the constant background,” Preprint (2018).Google Scholar
  61. 61.
    P. M. Santini, “The first rogue wave appearance in the rogue wave periodic Cauchy problem for the PT-symmetric NLS: Regular dynamics or blow up at finite time,” Preprint (2018).Google Scholar
  62. 62.
    H. C. Yuen and W. E. Ferguson, “Relationship between Benjamin-Feir instability and recurrence in the nonlinear Schrödinger equation,” Phys. Fluids, 21, 1275–1278 (1978).ADSCrossRefGoogle Scholar
  63. 63.
    B. M. Lake, H. C. Yuen, H. Rungaldier, and W. E. Ferguson, “Nonlinear deep-water waves: Theory and experiment. Part 2. Evolution of a continuous wave train,” J. Fluid Mech., 83, 49–74 (1977).ADSCrossRefGoogle Scholar
  64. 64.
    H. Yuen and B. Lake, “Nonlinear dynamics of deep-water gravity waves,” Adv. Appl. Mech., 22, 67–229 (1982).MathSciNetCrossRefzbMATHGoogle Scholar
  65. 65.
    N. N. Akhmediev, “Nonlinear physics: Déjà vu in optics,” Nature, 413, 267–268 (2001).ADSCrossRefGoogle Scholar
  66. 66.
    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).ADSCrossRefGoogle Scholar
  67. 67.
    E. A. Kuznetsov, “Fermi–Pasta–Ulam recurrence and modulation instability,” JETP Letters, 105, 125–129 (2017).ADSCrossRefGoogle Scholar
  68. 68.
    O. Kimmoun, H. C. Hsu, H. Branger, M. S. Li, Y. Y. Chen, C. Kharif, M. Onorato, E. J. R. Kelleher, B. Kibler, N. Akhmediev, and A. Chabchoub, “Modulation instability and phase-shifted Fermi–Pasta–Ulam recurrence,” Sci. Rep., 6, 28516 (2016).ADSCrossRefGoogle Scholar
  69. 69.
    A. Mussot, C. Naveau, M. Conforti, A. Kudlinski, P. Szriftgiser, F. Copie, and S. Trillo, “Fibre multiwave-mixing combs reveal the broken symmetry of Fermi–Pasta–Ulam recurrence,” Nature Photonics, 12, 303–308 (2018).ADSCrossRefGoogle Scholar
  70. 70.
    D. Pierangeli, M. Flammini, L. Zhang, G. Marcucci, A. J. Agranat, P. G. Grinevich, P. M. Santini, C. Conti, and E. DelRe, “Observation of exact Fermi–Pasta–Ulam recurrence,” Preprint (2018).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Landau Institute for Theoretical PhysicsRASChernogolovkaRussia
  2. 2.Lomonosov Moscow State UniversityMoscowRussia
  3. 3.Dipartimento di FisicaUniversità di Roma “La Sapienza,”RomeItaly
  4. 4.Istituto Nazionale di Fisica NucleareRomeItaly

Personalised recommendations