Abstract

Propagating and periodically amplified and filtered pulses are considered as resonant periodic orbits of the applied variational model. Launching conditions (initial width and chirp) resulting in selection of particular nonlinear modes are given by a Poincaré map analysis. The resonant periodic orbits and the corresponding nonlinear modes of the real system are shown to be destroyed under strong filtering. Application of Melnikov’s method provides estimates of filtering margin for specific nonlinear mode preservation. Direct simulations of the real system show that the nonlinear modes that are characterized by large-amplitude pulse-shape oscillations exhibit a remarkably low radiation emission.

© 2003 Optical Society of America

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  1. M. Nakazawa, Y. Kimura, and K. Suzuki, “Soliton amplification and transmission with Er3+-doped fibre repeater pumped by GaInAsP laser diode,” Electron. Lett. 25, 199–200 (1989).
    [CrossRef]
  2. A. Hasegawa and Y. Kodama, Solitons in Optical Communications (Clarendon, Oxford, UK, 1995).
  3. I. Gabitov, E. G. Shapiro, and S. K. Turitsyn, “Asymptotic breathing pulse in optical transmission systems with dispersion compensation,” Phys. Rev. E 55, 3624–3633 (1997).
    [CrossRef]
  4. S. K. Turitsyn, I. Gabitov, E. W. Laedke, V. K. Mezentsev, S. L. Musher, E. G. Shapiro, T. Schäfer, and K. H. Spatschek, “Variational approach to optical pulse propagation in dispersion-compensated transmission systems,” Opt. Commun. 151, 117–135 (1998).
    [CrossRef]
  5. H. Sugahara, K. Hiroki, I. Takashi, A. Maruta, and Y. Kodama, “Optimal dispersion management for a wavelength division multiplexed optical soliton transmission system,” J. Lightwave Technol. 17, 1547–1559 (1999).
    [CrossRef]
  6. B. A. Malomed, D. F. Parker, and N. F. Smyth, “Resonant shape oscillations and decay of a soliton in a periodically inhomogeneous nonlinear optical fiber,” Phys. Rev. E 48, 1418–1425 (1993).
    [CrossRef]
  7. B. A. Malomed, “Pulse propagation in a nonlinear optical fiber with periodically modulated dispersion: variational approach,” Opt. Commun. 136, 313–319 (1997).
    [CrossRef]
  8. T. I. Lakoba, J. Yang, D. J. Kaup, and B. A. Malomed, “Conditions for stationary propagations in the strong dispersion management limit,” Opt. Commun. 149, 366–375 (1998).
    [CrossRef]
  9. J. N. Kutz, P. Holmes, S. G. Evangelides, and J. P. Gordon, “Hamiltonian dynamics of dispersion-managed breathers,” J. Opt. Soc. Am. B 15, 87–96 (1998).
    [CrossRef]
  10. P. Holmes and J. N. Kutz, “Dynamics and bifurcations of a planar map modeling dispersion managed breathers,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 59, 1288–1302 (1999).
    [CrossRef]
  11. N. J. Smith and N. J. Doran, “Picosecond soliton transmis-sion using concatenated nonlinear optical loop-mirror intensity filters,” J. Opt. Soc. Am. B 12, 1117–1125 (1995).
    [CrossRef]
  12. R.-J. Essiambre and G. P. Agrawal, “Soliton communication beyond the average-soliton regime,” J. Opt. Soc. Am. B 12, 2420–2425 (1995).
    [CrossRef]
  13. W. Forysiak, N. J. Doran, F. M. Knox, and K. J. Blow, “Average soliton dynamics in strongly perturbed systems,” Opt. Commun. 117, 65–70 (1995).
    [CrossRef]
  14. Z. M. Liao, C. J. McKinstrie, and G. P. Agrawal, “Importance of prechirping in constant-dispersion fiber links with a large amplifier spacing,” J. Opt. Soc. Am. B 17, 514–518 (2000).
    [CrossRef]
  15. F. Kh. Abdullaev and J. G. Caputo, “Validation of the variational approach for chirped pulses in fibers with periodic dispersion,” Phys. Rev. E 58, 6637–6648 (1998).
    [CrossRef]
  16. R. Grimshaw, J. He, and B. A. Malomed, “Decay of a fundamental soliton in a periodically modulated nonlinear waveguide,” Phys. Scr. 53, 385–393 (1996).
    [CrossRef]
  17. D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 27, 3135–3145 (1983).
    [CrossRef]
  18. D. Anderson, M. Lisak, and T. Reichel, “Asymptotic propagation properties of pulses in a soliton-based optical-fiber communication system,” J. Opt. Soc. Am. B 5, 207–210 (1988).
    [CrossRef]
  19. B. Malomed, “Variational methods in nonlinear fiber optics and related fields,” Prog. Opt. 43, 69 (2002).
  20. F. Kh. Abdullaev, A. A. Abdumalikov, and B. B. Baizakov, “Stochastic instability of chirped optical solitons in media with periodic amplification,” Quantum Electron. 27, 171–175 (1997).
    [CrossRef]
  21. B. A. Malomed, “Resonant transmission of a chirped soliton in a long optical fiber with periodic amplification,” J. Opt. Soc. Am. B 13, 677–686 (1996).
    [CrossRef]
  22. Y. Kominis and K. Hizanidis, “Regular and chaotic dynamics of periodically amplified picosecond solitons,” J. Opt. Soc. Am. B 19, 1746–1758 (2002).
    [CrossRef]
  23. H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1980).
  24. A. J. Lichtenberg and M. A. Lieberman, Regular and Stochastic Motion (Springer-Verlag, New York, 1983).
  25. B. V. Chirikov, “A universal instability of many-dimensional oscillator systems,” Phys. Rep. 52, 263–379 (1979).
    [CrossRef]
  26. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer-Verlag, New York, 1983).
  27. Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763–915 (1989).
    [CrossRef]
  28. A. Bondeson, M. Lisak, and D. Andersen, “Soliton perturbations: a variational principle for the soliton parameters,” Phys. Scr. 20, 479–485 (1979).
    [CrossRef]
  29. B. A. Malomed, “Bound states of envelope solitons,” Phys. Rev. E 47, 2874–2880 (1993).
    [CrossRef]

2002

2000

1999

P. Holmes and J. N. Kutz, “Dynamics and bifurcations of a planar map modeling dispersion managed breathers,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 59, 1288–1302 (1999).
[CrossRef]

H. Sugahara, K. Hiroki, I. Takashi, A. Maruta, and Y. Kodama, “Optimal dispersion management for a wavelength division multiplexed optical soliton transmission system,” J. Lightwave Technol. 17, 1547–1559 (1999).
[CrossRef]

1998

S. K. Turitsyn, I. Gabitov, E. W. Laedke, V. K. Mezentsev, S. L. Musher, E. G. Shapiro, T. Schäfer, and K. H. Spatschek, “Variational approach to optical pulse propagation in dispersion-compensated transmission systems,” Opt. Commun. 151, 117–135 (1998).
[CrossRef]

T. I. Lakoba, J. Yang, D. J. Kaup, and B. A. Malomed, “Conditions for stationary propagations in the strong dispersion management limit,” Opt. Commun. 149, 366–375 (1998).
[CrossRef]

J. N. Kutz, P. Holmes, S. G. Evangelides, and J. P. Gordon, “Hamiltonian dynamics of dispersion-managed breathers,” J. Opt. Soc. Am. B 15, 87–96 (1998).
[CrossRef]

F. Kh. Abdullaev and J. G. Caputo, “Validation of the variational approach for chirped pulses in fibers with periodic dispersion,” Phys. Rev. E 58, 6637–6648 (1998).
[CrossRef]

1997

B. A. Malomed, “Pulse propagation in a nonlinear optical fiber with periodically modulated dispersion: variational approach,” Opt. Commun. 136, 313–319 (1997).
[CrossRef]

I. Gabitov, E. G. Shapiro, and S. K. Turitsyn, “Asymptotic breathing pulse in optical transmission systems with dispersion compensation,” Phys. Rev. E 55, 3624–3633 (1997).
[CrossRef]

F. Kh. Abdullaev, A. A. Abdumalikov, and B. B. Baizakov, “Stochastic instability of chirped optical solitons in media with periodic amplification,” Quantum Electron. 27, 171–175 (1997).
[CrossRef]

1996

B. A. Malomed, “Resonant transmission of a chirped soliton in a long optical fiber with periodic amplification,” J. Opt. Soc. Am. B 13, 677–686 (1996).
[CrossRef]

R. Grimshaw, J. He, and B. A. Malomed, “Decay of a fundamental soliton in a periodically modulated nonlinear waveguide,” Phys. Scr. 53, 385–393 (1996).
[CrossRef]

1995

1993

B. A. Malomed, D. F. Parker, and N. F. Smyth, “Resonant shape oscillations and decay of a soliton in a periodically inhomogeneous nonlinear optical fiber,” Phys. Rev. E 48, 1418–1425 (1993).
[CrossRef]

B. A. Malomed, “Bound states of envelope solitons,” Phys. Rev. E 47, 2874–2880 (1993).
[CrossRef]

1989

Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763–915 (1989).
[CrossRef]

M. Nakazawa, Y. Kimura, and K. Suzuki, “Soliton amplification and transmission with Er3+-doped fibre repeater pumped by GaInAsP laser diode,” Electron. Lett. 25, 199–200 (1989).
[CrossRef]

1988

1983

D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 27, 3135–3145 (1983).
[CrossRef]

1979

A. Bondeson, M. Lisak, and D. Andersen, “Soliton perturbations: a variational principle for the soliton parameters,” Phys. Scr. 20, 479–485 (1979).
[CrossRef]

B. V. Chirikov, “A universal instability of many-dimensional oscillator systems,” Phys. Rep. 52, 263–379 (1979).
[CrossRef]

Abdullaev, F. Kh.

F. Kh. Abdullaev and J. G. Caputo, “Validation of the variational approach for chirped pulses in fibers with periodic dispersion,” Phys. Rev. E 58, 6637–6648 (1998).
[CrossRef]

F. Kh. Abdullaev, A. A. Abdumalikov, and B. B. Baizakov, “Stochastic instability of chirped optical solitons in media with periodic amplification,” Quantum Electron. 27, 171–175 (1997).
[CrossRef]

Abdumalikov, A. A.

F. Kh. Abdullaev, A. A. Abdumalikov, and B. B. Baizakov, “Stochastic instability of chirped optical solitons in media with periodic amplification,” Quantum Electron. 27, 171–175 (1997).
[CrossRef]

Agrawal, G. P.

Andersen, D.

A. Bondeson, M. Lisak, and D. Andersen, “Soliton perturbations: a variational principle for the soliton parameters,” Phys. Scr. 20, 479–485 (1979).
[CrossRef]

Anderson, D.

Baizakov, B. B.

F. Kh. Abdullaev, A. A. Abdumalikov, and B. B. Baizakov, “Stochastic instability of chirped optical solitons in media with periodic amplification,” Quantum Electron. 27, 171–175 (1997).
[CrossRef]

Blow, K. J.

W. Forysiak, N. J. Doran, F. M. Knox, and K. J. Blow, “Average soliton dynamics in strongly perturbed systems,” Opt. Commun. 117, 65–70 (1995).
[CrossRef]

Bondeson, A.

A. Bondeson, M. Lisak, and D. Andersen, “Soliton perturbations: a variational principle for the soliton parameters,” Phys. Scr. 20, 479–485 (1979).
[CrossRef]

Caputo, J. G.

F. Kh. Abdullaev and J. G. Caputo, “Validation of the variational approach for chirped pulses in fibers with periodic dispersion,” Phys. Rev. E 58, 6637–6648 (1998).
[CrossRef]

Chirikov, B. V.

B. V. Chirikov, “A universal instability of many-dimensional oscillator systems,” Phys. Rep. 52, 263–379 (1979).
[CrossRef]

Doran, N. J.

W. Forysiak, N. J. Doran, F. M. Knox, and K. J. Blow, “Average soliton dynamics in strongly perturbed systems,” Opt. Commun. 117, 65–70 (1995).
[CrossRef]

N. J. Smith and N. J. Doran, “Picosecond soliton transmis-sion using concatenated nonlinear optical loop-mirror intensity filters,” J. Opt. Soc. Am. B 12, 1117–1125 (1995).
[CrossRef]

Essiambre, R.-J.

Evangelides, S. G.

Forysiak, W.

W. Forysiak, N. J. Doran, F. M. Knox, and K. J. Blow, “Average soliton dynamics in strongly perturbed systems,” Opt. Commun. 117, 65–70 (1995).
[CrossRef]

Gabitov, I.

S. K. Turitsyn, I. Gabitov, E. W. Laedke, V. K. Mezentsev, S. L. Musher, E. G. Shapiro, T. Schäfer, and K. H. Spatschek, “Variational approach to optical pulse propagation in dispersion-compensated transmission systems,” Opt. Commun. 151, 117–135 (1998).
[CrossRef]

I. Gabitov, E. G. Shapiro, and S. K. Turitsyn, “Asymptotic breathing pulse in optical transmission systems with dispersion compensation,” Phys. Rev. E 55, 3624–3633 (1997).
[CrossRef]

Gordon, J. P.

Grimshaw, R.

R. Grimshaw, J. He, and B. A. Malomed, “Decay of a fundamental soliton in a periodically modulated nonlinear waveguide,” Phys. Scr. 53, 385–393 (1996).
[CrossRef]

He, J.

R. Grimshaw, J. He, and B. A. Malomed, “Decay of a fundamental soliton in a periodically modulated nonlinear waveguide,” Phys. Scr. 53, 385–393 (1996).
[CrossRef]

Hiroki, K.

Hizanidis, K.

Holmes, P.

P. Holmes and J. N. Kutz, “Dynamics and bifurcations of a planar map modeling dispersion managed breathers,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 59, 1288–1302 (1999).
[CrossRef]

J. N. Kutz, P. Holmes, S. G. Evangelides, and J. P. Gordon, “Hamiltonian dynamics of dispersion-managed breathers,” J. Opt. Soc. Am. B 15, 87–96 (1998).
[CrossRef]

Kaup, D. J.

T. I. Lakoba, J. Yang, D. J. Kaup, and B. A. Malomed, “Conditions for stationary propagations in the strong dispersion management limit,” Opt. Commun. 149, 366–375 (1998).
[CrossRef]

Kimura, Y.

M. Nakazawa, Y. Kimura, and K. Suzuki, “Soliton amplification and transmission with Er3+-doped fibre repeater pumped by GaInAsP laser diode,” Electron. Lett. 25, 199–200 (1989).
[CrossRef]

Kivshar, Y. S.

Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763–915 (1989).
[CrossRef]

Knox, F. M.

W. Forysiak, N. J. Doran, F. M. Knox, and K. J. Blow, “Average soliton dynamics in strongly perturbed systems,” Opt. Commun. 117, 65–70 (1995).
[CrossRef]

Kodama, Y.

Kominis, Y.

Kutz, J. N.

P. Holmes and J. N. Kutz, “Dynamics and bifurcations of a planar map modeling dispersion managed breathers,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 59, 1288–1302 (1999).
[CrossRef]

J. N. Kutz, P. Holmes, S. G. Evangelides, and J. P. Gordon, “Hamiltonian dynamics of dispersion-managed breathers,” J. Opt. Soc. Am. B 15, 87–96 (1998).
[CrossRef]

Laedke, E. W.

S. K. Turitsyn, I. Gabitov, E. W. Laedke, V. K. Mezentsev, S. L. Musher, E. G. Shapiro, T. Schäfer, and K. H. Spatschek, “Variational approach to optical pulse propagation in dispersion-compensated transmission systems,” Opt. Commun. 151, 117–135 (1998).
[CrossRef]

Lakoba, T. I.

T. I. Lakoba, J. Yang, D. J. Kaup, and B. A. Malomed, “Conditions for stationary propagations in the strong dispersion management limit,” Opt. Commun. 149, 366–375 (1998).
[CrossRef]

Liao, Z. M.

Lisak, M.

D. Anderson, M. Lisak, and T. Reichel, “Asymptotic propagation properties of pulses in a soliton-based optical-fiber communication system,” J. Opt. Soc. Am. B 5, 207–210 (1988).
[CrossRef]

A. Bondeson, M. Lisak, and D. Andersen, “Soliton perturbations: a variational principle for the soliton parameters,” Phys. Scr. 20, 479–485 (1979).
[CrossRef]

Malomed, B.

B. Malomed, “Variational methods in nonlinear fiber optics and related fields,” Prog. Opt. 43, 69 (2002).

Malomed, B. A.

T. I. Lakoba, J. Yang, D. J. Kaup, and B. A. Malomed, “Conditions for stationary propagations in the strong dispersion management limit,” Opt. Commun. 149, 366–375 (1998).
[CrossRef]

B. A. Malomed, “Pulse propagation in a nonlinear optical fiber with periodically modulated dispersion: variational approach,” Opt. Commun. 136, 313–319 (1997).
[CrossRef]

B. A. Malomed, “Resonant transmission of a chirped soliton in a long optical fiber with periodic amplification,” J. Opt. Soc. Am. B 13, 677–686 (1996).
[CrossRef]

R. Grimshaw, J. He, and B. A. Malomed, “Decay of a fundamental soliton in a periodically modulated nonlinear waveguide,” Phys. Scr. 53, 385–393 (1996).
[CrossRef]

B. A. Malomed, “Bound states of envelope solitons,” Phys. Rev. E 47, 2874–2880 (1993).
[CrossRef]

B. A. Malomed, D. F. Parker, and N. F. Smyth, “Resonant shape oscillations and decay of a soliton in a periodically inhomogeneous nonlinear optical fiber,” Phys. Rev. E 48, 1418–1425 (1993).
[CrossRef]

Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763–915 (1989).
[CrossRef]

Maruta, A.

McKinstrie, C. J.

Mezentsev, V. K.

S. K. Turitsyn, I. Gabitov, E. W. Laedke, V. K. Mezentsev, S. L. Musher, E. G. Shapiro, T. Schäfer, and K. H. Spatschek, “Variational approach to optical pulse propagation in dispersion-compensated transmission systems,” Opt. Commun. 151, 117–135 (1998).
[CrossRef]

Musher, S. L.

S. K. Turitsyn, I. Gabitov, E. W. Laedke, V. K. Mezentsev, S. L. Musher, E. G. Shapiro, T. Schäfer, and K. H. Spatschek, “Variational approach to optical pulse propagation in dispersion-compensated transmission systems,” Opt. Commun. 151, 117–135 (1998).
[CrossRef]

Nakazawa, M.

M. Nakazawa, Y. Kimura, and K. Suzuki, “Soliton amplification and transmission with Er3+-doped fibre repeater pumped by GaInAsP laser diode,” Electron. Lett. 25, 199–200 (1989).
[CrossRef]

Parker, D. F.

B. A. Malomed, D. F. Parker, and N. F. Smyth, “Resonant shape oscillations and decay of a soliton in a periodically inhomogeneous nonlinear optical fiber,” Phys. Rev. E 48, 1418–1425 (1993).
[CrossRef]

Reichel, T.

Schäfer, T.

S. K. Turitsyn, I. Gabitov, E. W. Laedke, V. K. Mezentsev, S. L. Musher, E. G. Shapiro, T. Schäfer, and K. H. Spatschek, “Variational approach to optical pulse propagation in dispersion-compensated transmission systems,” Opt. Commun. 151, 117–135 (1998).
[CrossRef]

Shapiro, E. G.

S. K. Turitsyn, I. Gabitov, E. W. Laedke, V. K. Mezentsev, S. L. Musher, E. G. Shapiro, T. Schäfer, and K. H. Spatschek, “Variational approach to optical pulse propagation in dispersion-compensated transmission systems,” Opt. Commun. 151, 117–135 (1998).
[CrossRef]

I. Gabitov, E. G. Shapiro, and S. K. Turitsyn, “Asymptotic breathing pulse in optical transmission systems with dispersion compensation,” Phys. Rev. E 55, 3624–3633 (1997).
[CrossRef]

Smith, N. J.

Smyth, N. F.

B. A. Malomed, D. F. Parker, and N. F. Smyth, “Resonant shape oscillations and decay of a soliton in a periodically inhomogeneous nonlinear optical fiber,” Phys. Rev. E 48, 1418–1425 (1993).
[CrossRef]

Spatschek, K. H.

S. K. Turitsyn, I. Gabitov, E. W. Laedke, V. K. Mezentsev, S. L. Musher, E. G. Shapiro, T. Schäfer, and K. H. Spatschek, “Variational approach to optical pulse propagation in dispersion-compensated transmission systems,” Opt. Commun. 151, 117–135 (1998).
[CrossRef]

Sugahara, H.

Suzuki, K.

M. Nakazawa, Y. Kimura, and K. Suzuki, “Soliton amplification and transmission with Er3+-doped fibre repeater pumped by GaInAsP laser diode,” Electron. Lett. 25, 199–200 (1989).
[CrossRef]

Takashi, I.

Turitsyn, S. K.

S. K. Turitsyn, I. Gabitov, E. W. Laedke, V. K. Mezentsev, S. L. Musher, E. G. Shapiro, T. Schäfer, and K. H. Spatschek, “Variational approach to optical pulse propagation in dispersion-compensated transmission systems,” Opt. Commun. 151, 117–135 (1998).
[CrossRef]

I. Gabitov, E. G. Shapiro, and S. K. Turitsyn, “Asymptotic breathing pulse in optical transmission systems with dispersion compensation,” Phys. Rev. E 55, 3624–3633 (1997).
[CrossRef]

Yang, J.

T. I. Lakoba, J. Yang, D. J. Kaup, and B. A. Malomed, “Conditions for stationary propagations in the strong dispersion management limit,” Opt. Commun. 149, 366–375 (1998).
[CrossRef]

Electron. Lett.

M. Nakazawa, Y. Kimura, and K. Suzuki, “Soliton amplification and transmission with Er3+-doped fibre repeater pumped by GaInAsP laser diode,” Electron. Lett. 25, 199–200 (1989).
[CrossRef]

J. Lightwave Technol.

J. Opt. Soc. Am. B

Opt. Commun.

W. Forysiak, N. J. Doran, F. M. Knox, and K. J. Blow, “Average soliton dynamics in strongly perturbed systems,” Opt. Commun. 117, 65–70 (1995).
[CrossRef]

B. A. Malomed, “Pulse propagation in a nonlinear optical fiber with periodically modulated dispersion: variational approach,” Opt. Commun. 136, 313–319 (1997).
[CrossRef]

T. I. Lakoba, J. Yang, D. J. Kaup, and B. A. Malomed, “Conditions for stationary propagations in the strong dispersion management limit,” Opt. Commun. 149, 366–375 (1998).
[CrossRef]

S. K. Turitsyn, I. Gabitov, E. W. Laedke, V. K. Mezentsev, S. L. Musher, E. G. Shapiro, T. Schäfer, and K. H. Spatschek, “Variational approach to optical pulse propagation in dispersion-compensated transmission systems,” Opt. Commun. 151, 117–135 (1998).
[CrossRef]

Phys. Rep.

B. V. Chirikov, “A universal instability of many-dimensional oscillator systems,” Phys. Rep. 52, 263–379 (1979).
[CrossRef]

Phys. Rev. A

D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 27, 3135–3145 (1983).
[CrossRef]

Phys. Rev. E

B. A. Malomed, “Bound states of envelope solitons,” Phys. Rev. E 47, 2874–2880 (1993).
[CrossRef]

I. Gabitov, E. G. Shapiro, and S. K. Turitsyn, “Asymptotic breathing pulse in optical transmission systems with dispersion compensation,” Phys. Rev. E 55, 3624–3633 (1997).
[CrossRef]

B. A. Malomed, D. F. Parker, and N. F. Smyth, “Resonant shape oscillations and decay of a soliton in a periodically inhomogeneous nonlinear optical fiber,” Phys. Rev. E 48, 1418–1425 (1993).
[CrossRef]

F. Kh. Abdullaev and J. G. Caputo, “Validation of the variational approach for chirped pulses in fibers with periodic dispersion,” Phys. Rev. E 58, 6637–6648 (1998).
[CrossRef]

Phys. Scr.

R. Grimshaw, J. He, and B. A. Malomed, “Decay of a fundamental soliton in a periodically modulated nonlinear waveguide,” Phys. Scr. 53, 385–393 (1996).
[CrossRef]

A. Bondeson, M. Lisak, and D. Andersen, “Soliton perturbations: a variational principle for the soliton parameters,” Phys. Scr. 20, 479–485 (1979).
[CrossRef]

Prog. Opt.

B. Malomed, “Variational methods in nonlinear fiber optics and related fields,” Prog. Opt. 43, 69 (2002).

Quantum Electron.

F. Kh. Abdullaev, A. A. Abdumalikov, and B. B. Baizakov, “Stochastic instability of chirped optical solitons in media with periodic amplification,” Quantum Electron. 27, 171–175 (1997).
[CrossRef]

Rev. Mod. Phys.

Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763–915 (1989).
[CrossRef]

SIAM (Soc. Ind. Appl. Math.) J. Appl. Math.

P. Holmes and J. N. Kutz, “Dynamics and bifurcations of a planar map modeling dispersion managed breathers,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 59, 1288–1302 (1999).
[CrossRef]

Other

A. Hasegawa and Y. Kodama, Solitons in Optical Communications (Clarendon, Oxford, UK, 1995).

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer-Verlag, New York, 1983).

H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1980).

A. J. Lichtenberg and M. A. Lieberman, Regular and Stochastic Motion (Springer-Verlag, New York, 1983).

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Figures (9)

Fig. 1
Fig. 1

Bifurcation curves for subharmonic resonant orbits for (a) ω0=Ω/2, (b) ω0=Ω.

Fig. 2
Fig. 2

Poincaré surface of section of the variational model corresponding to a pulse propagating in a transmission link with the following pulse width, amplification period, filtering parameter and pulse energy: (a) ts=4 ps, d=100 km, γ1=10-4, with N0 chosen such that ω0=Ω; (b) ts=8 ps, d=100 km, γ1=10-3, with N0 chosen such that ω0=Ω/2; (c) ts=8 ps, d=100 km, γ1=10-3, with N0 chosen such that ω0=Ω; (d) the same set of parameters as in (c) except γ1=7×10-3.

Fig. 3
Fig. 3

Pulse propagation for initial width and chirp close to the first subharmonic resonance corresponding to point A of Fig. 2(c): (a) direct integration of the perturbed NLS equation; (b) amplitude oscillations as obtained from direct integration (solid curve) and the variational model (dotted curve).

Fig. 4
Fig. 4

Pulse propagation for initial width and chirp close to the second subharmonic resonance corresponding to point B of Fig. 2(c): (a) direct integration of the perturbed NLS equation; (b) amplitude oscillations as obtained from direct integration (solid curve) and the variational model (dotted curve).

Fig. 5
Fig. 5

Pulse propagation for initial width and chirp close to the third subharmonic resonance corresponding to point C of Fig. 2(c): (a) direct integration of the perturbed NLS equation; (b) amplitude oscillations as obtained from direct integration (solid curve) and the variational model (dotted curve).

Fig. 6
Fig. 6

Pulse propagation for initial width and chirp lying far from subharmonic resonances corresponding to point D of Fig. 2(c), obtained by direct integration of the perturbed NLS equation.

Fig. 7
Fig. 7

Pulse propagation for initial width and chirp corresponding to point A of Fig. 2(d), obtained by direct integration of the perturbed NLS equation.

Fig. 8
Fig. 8

Pulse propagation for initial width and chirp corresponding to point B of Fig. 2(d), obtained by direct integration of the perturbed NLS equation.

Fig. 9
Fig. 9

Pulse propagation for initial width and chirp corresponding to point C of Fig. 2(d): (a) direct integration of the perturbed NLS equation; (b) drastic change in pulse shape shown in detail.

Equations (27)

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i qZ+122qT2+|q|2q=iγ0(Z)q+iγ12qT2.
Z0=6.07×102ts2 [ps]λ2 [μm] D [(ps/nm)/km],
γ0(Z)=-Γ0+Γ1n=-+δ(Z-Zan),
Γ0=7×10-2ts2 [ps] δ [dB/km]λ2[μm] D [(ps/nm)/km].
γ¯0=13 γ1N4,
N2=12-+|q(T)|2dT.
i uZ+122uT2+exp[2Γ˜(Z)]|u|2u=iγ¯0u+iγ12qT2.
u(Z, T)=A(Z)sechTα(Z)exp[iϕ0(Z)T2],
d2αdZ2=4π2α3-4N2exp[Δ(Z)]π2α2-2λγ1N4πdαdZ,
dNdZ=2Nγ¯0-13 γ1N4-23 π2γ1N-5α-1b2,
Δ(Z)=4π Γ˜(Z).
ϕ0=12d(ln α)dZ-4γ1π2α2.
H0α, dαdZ=12dαdZ2+2π2α2-4N02π2α=E,
ω(E)=π22N02 (-E)3/2.
α(Z)=b(1-e0cos ξ),ωZ=ξ-e0sin ξ,
e0=1-π2|E|2N041/2,b=2N02π2|E|,
mω(E)-nΩ=0,
x˙=y,
y˙=4π2x3-4N02π2x2-4N02π2x2 Δ(Z)-2λγ1N04π y,
Mm/n(Z0; ts, d, γ1, N0)
=-0mTy4N02π2x2 Δ(Z+Z0)+2λγ1N04π ydZ,
Δ(Z)=4π2 Γ1n0n-12iexpi 2πnZa Z.
Mm/n(Z0; ts, d, γ1, N0)
=-32π3N02Γ1bmn mk=1Jk m/n(k (m/n)e0mn)cos(kΩZ0)-4λγ1N04nωmn{1-[1-(e0mn)2]1/2}
Rm/n(ts, d, N0)
=8Γ1λπ3N02Ω(m/n)2bmn(1-1-(e0mn)2)×k=1Jk m/n(k(m/n)e0mn),
e0mn=0

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