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Theoretical and Mathematical Physics

, Volume 120, Issue 2, pp 997–1008 | Cite as

Modulation instability of soliton trains in fiber communication systems

  • E. A. Kuznetsov
  • M. D. Spector
Article

Abstract

The linear stability problem for a soliton train described by the nonlinear Schrödinger equation is exactly solved using a linearization of the Zakharov-Shabat dressing procedure. This problem is reduced to finding a compatible solution of two linear equations. This approach allows the growth rate of the soliton lattice instability and the corresponding eigenfunctions to be found explicitly in a purely algebraic way. The growth rate can be expressed in terms of elliptic functions. Analysis of the dispersion relations and eigerfunctions shows that the solution, which has the form of a soliton train, is stable for defocusing media and unstable for focusing media with arbitrary parameters. Possible applications of the stability results to fiber communication systems are discussed.

Keywords

Soliton Spectral Parameter Elliptic Function Modulation Instability Dark Soliton 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    A. Hasegawa and F. Tappet,Appl. Phys. Lett.,23, 142 (1973).CrossRefADSGoogle Scholar
  2. 2.
    V. E. Zakharov and A. B. Shabat,JETP,61, 62 (1972).MathSciNetADSGoogle Scholar
  3. 3.
    C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. B. Miura,Phys. Rev. Lett.,19, 1095 (1967).zbMATHCrossRefADSGoogle Scholar
  4. 4.
    L. F. Mollenauer, R. H. Stolen, and M. N. Islam,Opt. Lett.,10, 229 (1985).ADSCrossRefGoogle Scholar
  5. 5.
    L. F. Mollenauer, E. Lichtman, M. J. Neibelt, and G. T. Harvey,Electron. Lett.,29, 910 (1993).Google Scholar
  6. 6.
    L. F. Mollenauer, J. P. Gordon, and P. V. Mamyshev, “Solitons in high bit-rate, long-distance transmission,” in:Optical Fiber Telecommunications (I. P. Karamzin and T. L. Koch, eds.), Vol. 3 A, Chap. 12, Acad. Press, San Diego (1997), p. 373.Google Scholar
  7. 7.
    T. V. Benjamin and J. E. Feir,J. Fluid Mech.,27, 417 (1967).zbMATHCrossRefADSGoogle Scholar
  8. 8.
    V. E. Zakharov, “Collapse and self-focusing of the Langmure waves,” in:Handbook of Plasma Physics (A. Galeev and R. Sudan, eds.), Vol. 3, Elsevier, Amsterdam (1984), p. 81.Google Scholar
  9. 9.
    M. D. Spector, 1988 (unpublished).Google Scholar
  10. 10.
    E. A. Kuznetsov, M. D. Spector, and G. E. Falkovich,Physica D,10, 379 (1984).zbMATHCrossRefMathSciNetADSGoogle Scholar
  11. 11.
    V. E. Zakharov and A. B. Shabat,Funct. Anal. Appl.,13, 166 (1980).Google Scholar
  12. 12.
    V. E. Zakharov and A. B. Shabat,JETP,37, 823 (1973).ADSGoogle Scholar
  13. 13.
    E. T. Whittaker and G. N. Watson,A Course of Modern Analysis, Cambridge Univ. Press, Cambridge (1927).zbMATHGoogle Scholar
  14. 14.
    E. A. Kuznetsov and A. V. Mikhailov,JETP,40, 855 (1974).MathSciNetADSGoogle Scholar
  15. 15.
    E. A. Kuznetsov,Dokl. Akad. Nauk SSSR,326, 575 (1977).Google Scholar
  16. 16.
    I. S. Gradshtein and I. M. Ryzhik,Tables of Integrals, Series, and Products, Acad. Press, New York (1980).Google Scholar

Copyright information

© Kluwer Academic/Plenum Publishers 1999

Authors and Affiliations

  • E. A. Kuznetsov
    • 1
  • M. D. Spector
    • 2
  1. 1.Landau Institute for Theoretical PhysicsMoscowRussia
  2. 2.Department of Applied MathematicsColorado University in BoulderBoulderUSA

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