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  • Letter
  • Open Access

Improved effective linearization of nonlinear Schrödinger waves by increasing nonlinearity

Katelyn Plaisier Leisman1, Douglas Zhou2, J. W. Banks3, Gregor Kovačič3,*, and David Cai2,4

  • 1Northwestern University, Department of Engineering Sciences and Applied Mathematics, Evanston, Illinois 60208, USA
  • 2School of Mathematical Sciences, MOE-LSC, CMA-Shanghai, and Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai 200240, People's Republic of China
  • 3Rensselaer Polytechnic Institute, Department of Mathematical Sciences, Troy, New York 12180, USA
  • 4Courant Institute of Mathematical Sciences, New York University, New York, New York 10012, USA

  • *kovacg@rpi.edu
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Phys. Rev. Research 4, L012009 – Published 31 January, 2022

DOI: https://doi.org/10.1103/PhysRevResearch.4.L012009

Abstract

From among the waves whose dynamics are governed by the nonlinear Schrödinger equation, we find a robust, spatiotemporally disordered family, in which waves initialized with increasing amplitudes, on average, over long timescales, effectively evolve as ever more weakly coupled collections of plane waves. In particular, the relative amount of energy contained in their coupling decays to zero with increasing wave amplitude.

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References (44)

  1. J. Satsuma and N. Yajima, B. Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media, Prog. Theor. Phys. Suppl. 55, 284 (1974).
  2. D. J. Benney and A. C. Newell, The propagation of nonlinear wave envelopes, J. Math. Phys. 46, 133 (1967).
  3. V. E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid, J. Appl. Mech. Tech. Phys. 9, 190 (1968).
  4. V. E. Zakharov and A. B. Shabat, Exact theory of two-dimensional self-focussing and one-dimensional self-modulating waves in nonlinear media, Soviet J. Expt. Theor. Phys. 34, 62 (1972).
  5. A. Hasegawa and F. Tappert, Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers, Appl. Phys. Lett. 23, 142 (1973).
  6. M. Onorato, A. R. Osborne, M. Serio, and L. Cavaleri, Modulational instability and non-Gaussian statistics in experimental random water-wave trains, Phys. Fluids 17, 078101 (2005).
  7. M. Onorato, A. Osborne, M. Serio, L. Cavaleri, C. Brandini, and C. Stansberg, Extreme waves, modulational instability and second order theory: Wave flume experiments on irregular waves, Eur. J. Mech. B Fluids 25, 586 (2006).
  8. M. Onorato, S. Residori, U. Bortolozzo, A. Montina, and F. Arecchi, Rogue waves and their generating mechanisms in different physical contexts, Phys. Rep. 528, 47 (2013).
  9. P. Suret, R. E. Koussaifi, A. Tikan, C. Evain, S. Randoux, C. Szwaj, and S. Bielawski, Single-shot observation of optical rogue waves in integrable turbulence using time microscopy, Nat. Commun. 7, 13136 (2016).
  10. A. Tikan, F. Bonnefoy, G. Ducrozet, G. Prabhudesai, G. Michel, A. Cazaubiel, Éric Falcon, F. Copie, S. Randoux, and P. Suret, Nonlinear dispersion relation in integrable turbulence, arXiv:2107.00575.
  11. M. B. Erdoğan and V. Zharnitsky, Quasi-linear dynamics in nonlinear Schrödinger equation with periodic boundary conditions, Commun. Math. Phys. 281, 655 (2008).
  12. W. Lee, G. Kovačič, and D. Cai, Renormalized Resonance Quartets in Dispersive Wave Turbulence, Phys. Rev. Lett. 103, 024502 (2009).
  13. K. P. Leisman, D. Zhou, J. W. Banks, G. Kovačič, and D. Cai, Effective dispersion in the focusing nonlinear schrödinger equation, Phys. Rev. E 100, 022215 (2019).
  14. A. M. Yaglom, Correlation Theory of Stationary and Related Random Functions. Volume I: Basic Results (Springer-Verlag, New York, 1987).
  15. H. Hasimoto and H. Ono, Nonlinear modulation of gravity waves, J. Phys. Soc. Jpn. 33, 805 (1972).
  16. J. Weideman and B. Herbst, Split-step methods for the solution of the nonlinear Schrödinger equation, SIAM J. Numer. Anal. 23, 485 (1986).
  17. S. Filip, A. Javeed, and L. Trefethen, Smooth random functions, random ODEs, and Gaussian processes, SIAM Rev. 61, 185 (2019).
  18. Y. Wakata, Frequency-wavenumber spectra of equatorial waves detected from satellite altimeter data, J. Oceanography 63, 483 (2007).
  19. J. T. Farrar, Observations of the dispersion characteristics and meridional sea level structure of equatorial waves in the Pacific ocean, J. Phys. Oceanography 38, 1669 (2008).
  20. T. Shinoda, G. N. Kiladis, and P. E. Roundy, Statistical representation of equatorial waves and tropical instability waves in the pacific ocean, Atmos. Res. 94, 37 (2009).
  21. T. Shinoda, Observed dispersion relation of Yanai waves and 17-day tropical instability waves in the pacific ocean, SOLA 6, 17 (2010).
  22. B. Fornberg and G. B. Whitham, A numerical and theoretical study of certain nonlinear wave phenomena, Philos. Trans. R. Soc. A - Math. Phys. Eng. Sci. 289, 373 (1978).
  23. V. E. Zakharov and S. V. Manakov, On the complete integrability of a nonlinear schrödinger equation, Theor. Math. Phys. 19, 551 (1974).
  24. V. Zakharov, V. L'vov, and G. Falkovich, Kolmogorov Spectra of Turbulence I–Wave Turbulence (Springer, Berlin, 1992).
  25. K. Ohe, K. Asano, and S. Takeda, Nonlinear dispersion relation of finite amplitude ionisation waves. II. Experiment, J. Phys. D 14, 1023 (1981).
  26. T. H. C. Herbers, S. Elgar, N. A. Sarap, and R. T. Guza, Nonlinear dispersion of surface gravity waves in shallow water, J. Phys. Oceanogr. 32, 1181 (2002).
  27. B. Gershgorin, Y. V. Lvov, and D. Cai, Renormalized waves and discrete breathers in β-Fermi-Pasta-Ulam chains, Phys. Rev. Lett. 95, 264302 (2005).
  28. B. Gershgorin, Y. V. Lvov, and D. Cai, Interactions of renormalized waves in thermalized Fermi-Pasta-Ulam chains, Phys. Rev. E 75, 046603 (2007).
  29. E. Herbert, N. Mordant, and E. Falcon, Observation of the nonlinear dispersion relation and spatial statistics of wave turbulence on the surface of a fluid, Phys. Rev. Lett. 105, 144502 (2010).
  30. W. Lee, G. Kovačič, and D. Cai, Generation of dispersion in nondispersive nonlinear waves in thermal equilibrium, Proc. Natl. Acad. Sci. USA 110, 3237 (2013).
  31. Shi-xiao W. Jiang, H.-H. Lu, D. Zhou, and D. Cai, Renormalized dispersion relations of β-Fermi-Pasta-Ulam chains in equilibrium and nonequilibrium states, Phys. Rev. E 90, 032925 (2014).
  32. K. L. Manktelow, M. J. Leamy, and M. Ruzzene, Analysis and experimental estimation of nonlinear dispersion in a periodic string, J. Vibrat. Acoust. 136, 031016 (2014).
  33. S. Jiang, H. Lu, D. Zhou, and D. Cai, Stochastic linearization of turbulent dynamics of dispersive waves in equilibrium and non-equilibrium state, New J. Phys. 18, 083028 (2016).
  34. X. Jiang, H. Yuan, and X. Sun, Nonlinear plasmonic dispersion and coupling analysis in the symmetric graphene sheets waveguide, Sci. Rep. 6, 39309 (2016).
  35. G. Agrawal, Nonlinear Fiber Optics (Academic, Waltham, MA, 2013).
  36. I. R. Gabitov and S. K. Turitsyn, Averaged pulse dynamics in a cascaded transmission system with passive dispersion compensation, Opt. Lett. 21, 327 (1996).
  37. I. Gabitov and S. K. Turitsyn, Breathing solitons in optical fiber links, J. Expt. Theor. Phys. Lett. 63, 861 (1996).
  38. M. J. Ablowitz and G. Biondini, Multiscale pulse dynamics in communication systems with strong dispersion management, Opt. Lett. 23, 1668 (1998).
  39. M. A. Newhouse, D. L. Weidman, and D. W. Hall, Enhanced-nonlinearity single-mode lead silicate optical fiber, Opt. Lett. 15, 1185 (1990).
  40. M. Asobe, T. Kanamori, and K. Kubodera, Ultrafast all-optical switching using highly nonlinear chalcogenide glass fiber, IEEE Photon. Technol. Lett. 4, 362 (1992).
  41. K. Kikuchi, All-optical signal processing using fiber nonlinearity, in The 15th Annual Meeting of the IEEE Lasers and Electro-Optics Society (IEEE, 2002), Vol. 2, pp. 428–429.
  42. T. Hasegawa, T. Nagashima, and N. Sugimoto, Dispersion properties of Bi2O3-based high nonlinear optical fiber, J. Ceram. Soc. Jpn. 114, 224 (2006).
  43. N. Sugimoto, Fabrication and characteristics of bismuth oxide based highly nonlinear fiber, J. Ceram. Soc. Jpn. 116, 1028 (2008).
  44. S. Saha, A. Dutta, C. DeVault, B. T. Diroll, R. D. Schaller, Z. Kudyshev, X. Xu, A. Kildishev, V. M. Shalaev, and A. Boltasseva, Extraordinarily large permittivity modulation in zinc oxide for dynamic nanophotonics, Mater. Today 43, 27 (2021).

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Phys. Rev. Research 4, L012009– Published 31 January, 2022
  • Received 19 September 2019
  • Revised 7 October 2021
  • Accepted 28 November 2021
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