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Quasi-analytical Perturbation Analysis of the Generalized Nonlinear Schrödinger Equation

  • J. Bonetti
  • S. M. Hernandez
  • P. I. FierensEmail author
  • E. Temprana
  • D. F. Grosz
Conference paper
Part of the Understanding Complex Systems book series (UCS)

Abstract

The Generalized Nonlinear Schrödinger Equation (GNLSE) finds several applications, especially in describing pulse propagation in nonlinear fiber optics. A well-known and thoroughly studied phenomenon in nonlinear wave propagation is that of modulation instability (MI). MI is approached as a weak perturbation to a pump and the analysis is based on preserving those terms linear on the perturbation and disregarding higher-order terms. In this sense, the linear MI analysis is relevant to the understanding of the onset of many other nonlinear phenomena, but its application is limited to the evolution of the perturbation over short distances. In this work, we propose quasi-analytical approximations to the propagation of a perturbation consisting of additive white noise that go beyond the linear modulation instability analysis. Moreover, we show these approximations to be in excellent agreement with numerical simulations and experimental measurements.

Notes

Acknowledgments

We gratefully acknowledge S. Radic for hosting J. B.’s research stay at the Photonic Systems Group, UCSD, financial support from project PIP 2015, CONICET, Argentina, and from ONR Global through the Visiting Scientists Program.

References

  1. 1.
    G. Agrawal, Nonlinear Fiber Optics, 5th edn. Optics and Photonics (Academic, New York, 2012)Google Scholar
  2. 2.
    V.E. Zakharov, Sov. Phys. JETP 35, 908 (1972)Google Scholar
  3. 3.
    M.A. Ablowitz, P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering (Cambridge University Press, Cambridge, 1991)Google Scholar
  4. 4.
    N. Akhmediev, V. Korneev, Theor. Math. Phys. 69(2), 1089 (1986)CrossRefGoogle Scholar
  5. 5.
    J.M. Dudley, G. Genty, F. Dias, B. Kibler, N. Akhmediev, Opt. Express 17(24), 21497 (2009).  https://doi.org/10.1364/OE.17.021497CrossRefGoogle Scholar
  6. 6.
    N. Akhmediev, J.M. Soto-Crespo, A. Ankiewicz, Phys. Rev. A 80, 043818 (2009).  https://doi.org/10.1103/PhysRevA.80.043818CrossRefGoogle Scholar
  7. 7.
    N. Akhmediev, A. Ankiewicz, M. Taki, Phys. Lett. A 373(6), 675 (2009).  https://doi.org/10.1016/j.physleta.2008.12.036CrossRefGoogle Scholar
  8. 8.
    A. Ankiewicz, J.M. Soto-Crespo, M.A. Chowdhury, N. Akhmediev, J. Opt. Soc. Am. B 30(1), 87 (2013).  https://doi.org/10.1364/JOSAB.30.000087CrossRefGoogle Scholar
  9. 9.
    A. Ankiewicz, N. Akhmediev, Phys. Lett. A 378(4), 358 (2014).  https://doi.org/10.1016/j.physleta.2013.11.031MathSciNetCrossRefGoogle Scholar
  10. 10.
    A. Ankiewicz, Y. Wang, S. Wabnitz, N. Akhmediev, Phys. Rev. E 89, 012907 (2014).  https://doi.org/10.1103/PhysRevE.89.012907CrossRefGoogle Scholar
  11. 11.
    A. Ankiewicz, D.J. Kedziora, A. Chowdury, U. Bandelow, N. Akhmediev, Phys. Rev. E 93, 012206 (2016).  https://doi.org/10.1103/PhysRevE.93.012206MathSciNetCrossRefGoogle Scholar
  12. 12.
    J. Hult, J. Light. Technol. 25(12), 3770 (2007).  https://doi.org/10.1109/JLT.2007.909373CrossRefGoogle Scholar
  13. 13.
    T.B. Benjamin, J.E. Feir, J. Fluid Mech. 27, 417 (1967).  https://doi.org/10.1017/S002211206700045XCrossRefGoogle Scholar
  14. 14.
    A. Hasegawa, Phys. Rev. Lett. 24, 1165 (1970).  https://doi.org/10.1103/PhysRevLett.24.1165CrossRefGoogle Scholar
  15. 15.
    V. Zakharov, A. Shabat, Sov. Phys. JETP 34, 62 (1972)Google Scholar
  16. 16.
    A. Hasegawa, W. Brinkman, IEEE J. Quantum Electron. 16(7), 694 (1980).  https://doi.org/10.1109/JQE.1980.1070554CrossRefGoogle Scholar
  17. 17.
    P.A.E.M. Janssen, Phys. Fluids 24(1), 23 (1981).  https://doi.org/10.1063/1.863242CrossRefGoogle Scholar
  18. 18.
    D. Anderson, M. Lisak, Opt. Lett. 9(10), 468 (1984).  https://doi.org/10.1364/OL.9.000468CrossRefGoogle Scholar
  19. 19.
    P.K. Shukla, J.J. Rasmussen, Opt. Lett. 11(3), 171 (1986).  https://doi.org/10.1364/OL.11.000171CrossRefGoogle Scholar
  20. 20.
    K. Tai, A. Hasegawa, A. Tomita, Phys. Rev. Lett. 56, 135 (1986).  https://doi.org/10.1103/PhysRevLett.56.135CrossRefGoogle Scholar
  21. 21.
    M.J. Potasek, Opt. Lett. 12(11), 921 (1987).  https://doi.org/10.1364/OL.12.000921CrossRefGoogle Scholar
  22. 22.
    M. Erkintalo, K. Hammani, B. Kibler, C. Finot, N. Akhmediev, J.M. Dudley, G. Genty, Phys. Rev. Lett. 107, 253901 (2011).  https://doi.org/10.1103/PhysRevLett.107.253901CrossRefGoogle Scholar
  23. 23.
    D. Solli, G. Herink, B. Jalali, C. Ropers, Nat. Photonics 6(7), 463 (2012).  https://doi.org/10.1038/nphoton.2012.126
  24. 24.
    D. Grosz, C. Mazzali, S. Celaschi, A. Paradisi, H. Fragnito, IEEE Photonics Technol. Lett. 11(3), 379 (1999).  https://doi.org/10.1109/68.748242CrossRefGoogle Scholar
  25. 25.
    D. Grosz, J.C. Boggio, H. Fragnito, Opt. Commun. 171(1–3), 53 (1999).  https://doi.org/10.1016/S0030-4018(99)00494-0
  26. 26.
    K. Hammani, B. Wetzel, B. Kibler, J. Fatome, C. Finot, G. Millot, N. Akhmediev, J.M. Dudley, Opt. Lett. 36(11), 2140 (2011).  https://doi.org/10.1364/OL.36.002140CrossRefGoogle Scholar
  27. 27.
    S.T. Sørensen, C. Larsen, U. Møller, P.M. Moselund, C.L. Thomsen, O. Bang, J. Opt. Soc. Am. B 29(10), 2875 (2012).  https://doi.org/10.1364/JOSAB.29.002875CrossRefGoogle Scholar
  28. 28.
    J.M. Soto-Crespo, A. Ankiewicz, N. Devine, N. Akhmediev, J. Opt. Soc. Am. B 29(8), 1930 (2012).  https://doi.org/10.1364/JOSAB.29.001930CrossRefGoogle Scholar
  29. 29.
    V.E. Zakharov, A.A. Gelash, Phys. Rev. Lett. 111, 054101 (2013).  https://doi.org/10.1103/PhysRevLett.111.054101CrossRefGoogle Scholar
  30. 30.
    P. Béjot, B. Kibler, E. Hertz, B. Lavorel, O. Faucher, Phys. Rev. A 83, 013830 (2011).  https://doi.org/10.1103/PhysRevA.83.013830CrossRefGoogle Scholar
  31. 31.
    S.M. Hernandez, P.I. Fierens, J. Bonetti, A.D. Sánchez, D.F. Grosz, IEEE Photonics J. 9(5), 1 (2017).  https://doi.org/10.1109/JPHOT.2017.2754984
  32. 32.
    P. Fierens, S. Hernandez, J. Bonetti, D. Grosz, in Proceedings of the 4th International Conference on Applications in Nonlinear Dynamics (ICAND 2016), ed. by V. In, P. Longhini, A. Palacios (Springer, Berlin, 2016), pp. 265–276.  https://doi.org/10.1007/978-3-319-52621-8_23
  33. 33.
    J. Bonetti, S.M. Hernandez, P.I. Fierens, D.F. Grosz, Phys. Rev. A 94, 033826 (2016).  https://doi.org/10.1103/PhysRevA.94.033826CrossRefGoogle Scholar
  34. 34.
    V. Zakharov, F. Dias, A. Pushkarev, Phys. Rep. 398(1), 1 (2004).  https://doi.org/10.1016/j.physrep.2004.04.002MathSciNetCrossRefGoogle Scholar
  35. 35.
    A. Picozzi, S. Pitois, G. Millot, Phys. Rev. Lett. 101, 093901 (2008).  https://doi.org/10.1103/PhysRevLett.101.093901CrossRefGoogle Scholar
  36. 36.
    A. Picozzi, S. Rica, Opt. Commun. 285(24), 5440 (2012).  https://doi.org/10.1016/j.optcom.2012.07.081
  37. 37.
    A. Picozzi, J. Garnier, T. Hansson, P. Suret, S. Randoux, G. Millot, D. Christodoulides, Phys. Rep. 542(1), 1 (2014).  https://doi.org/10.1016/j.physrep.2014.03.002
  38. 38.
    J.M. Soto-Crespo, N. Devine, N. Akhmediev, Phys. Rev. Lett. 116, 103901 (2016).  https://doi.org/10.1103/PhysRevLett.116.103901CrossRefGoogle Scholar
  39. 39.
    B.P.P. Kuo, J.M. Fini, L. Grüner-Nielsen, S. Radic, Opt. Express 20(17), 18611 (2012).  https://doi.org/10.1364/OE.20.018611CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • J. Bonetti
    • 1
    • 2
  • S. M. Hernandez
    • 1
  • P. I. Fierens
    • 2
    • 3
    Email author
  • E. Temprana
    • 4
  • D. F. Grosz
    • 1
    • 3
  1. 1.Instituto Balseiro (IB)Río NegroArgentina
  2. 2.Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET)Buenos AiresArgentina
  3. 3.Instituto Tecnológico de Buenos Aires (ITBA)Buenos AiresArgentina
  4. 4.Department of Electrical and Computer EngineeringUniversity of California San DiegoSan DiegoUSA

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