Abstract

Chirped-pulse propagation under periodic amplification is considered on the basis of the variational method, and the resulting pulse-shape chaotic oscillations are studied. The system of equations governing the evolution of the parameter functions is nonintegrable and is solved by the canonical perturbation method and the construction of local approximate invariants embracing all the essential features of the phase-space dynamics. The latter provide useful guidelines for choosing the appropriate launching-pulse width and chirp for stable propagation for each specific transmission-link configuration. This fact is supported by comparison of the analytic results with the respective numerical ones of the exact dynamical system obtained by the variational method and by the direct integration of the nonlinear Schrödinger equation as well. The structure of the chaotic layer between the two distinct modes of behavior of a propagating pulse, namely, breathing and spreading/decaying, is also investigated qualitatively by utilizing Melnikov’s method. Examples from technologically realistic configurations are given for 4–14-ps pulses and for amplification periods of 40–100 km.

© 2002 Optical Society of America

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  5. 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).
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  7. 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]
  8. B. A. Malomed, “Pulse propagation in a nonlinear optical fiber with periodically modulated dispersion: variational approach,” Opt. Commun. 136, 313–319 (1997).
    [CrossRef]
  9. 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).
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    [CrossRef]
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2000 (1)

1999 (2)

1998 (5)

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]

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]

F. Diacu and D. Şelaru, “Chaos in the Gyldén problem,” J. Math. Phys. 39, 6537–6546 (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]

1997 (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]

B. A. Malomed, “Pulse propagation in a nonlinear optical fiber with periodically modulated dispersion: variational approach,” Opt. Commun. 136, 313–319 (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 (3)

1995 (5)

R.-J. Essiambre and G. P. Agrawal, “Soliton communication beyond the average-soliton regime,” J. Opt. Soc. Am. B 12, 2420–2425 (1995).
[CrossRef]

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

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]

E. A. Kuznetsov, A. V. Mikhailov, and I. A. Shimokhin, “Nonlinear interaction of solitons and radiation,” Physica D 87, 201–215 (1995).
[CrossRef]

R. G. Bauer and L. A. Melnikov, “Multi-soliton fission and quasi-periodicity in a fiber with a periodically modulated core diameter,” Opt. Commun. 115, 190–198 (1995).
[CrossRef]

1994 (1)

G. M. Zaslavsky, “The width of exponentially narrow stochastic layers,” Chaos 4, 589–591 (1994).
[CrossRef] [PubMed]

1993 (1)

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]

1992 (1)

1989 (1)

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

1988 (1)

1983 (1)

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

1979 (1)

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

1975 (1)

J. B. Taylor and E. W. Laing, “Invariant for a particle interacting with an electrostatic wave in a magnetic field,” Phys. Rev. Lett. 35, 1306–1307 (1975).
[CrossRef]

1884 (1)

H. Gyldén, “Die Bahnbewegung in einem Systeme von zwei Körpern in dem Falle dass die Massen Veränderungen unterworfen sind,” Astron. Nachr. 109, 1–6 (1884).
[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.

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]

Bauer, R. G.

R. G. Bauer and L. A. Melnikov, “Multi-soliton fission and quasi-periodicity in a fiber with a periodically modulated core diameter,” Opt. Commun. 115, 190–198 (1995).
[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]

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]

Diacu, F.

F. Diacu and D. Şelaru, “Chaos in the Gyldén problem,” J. Math. Phys. 39, 6537–6546 (1998).
[CrossRef]

Doran, N. J.

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

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]

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]

I. Gabitov and S. K. Turitsyn, “Averaged pulse dynamics in a cascaded transmission system with passive dispersion compensation,” Opt. Lett. 21, 327–329 (1996).
[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]

Gyldén, H.

H. Gyldén, “Die Bahnbewegung in einem Systeme von zwei Körpern in dem Falle dass die Massen Veränderungen unterworfen sind,” Astron. Nachr. 109, 1–6 (1884).
[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.

Holmes, P.

P. Holmes and J. N. Kutz, “Dynamics and bifurcations of a planar map modeling dispersion managed breathers,” SIAM 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 fiber repeater pumped by GaInAsP laser diode,” Electron. Lett. 25, 199–200 (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.

Kutz, J. N.

P. Holmes and J. N. Kutz, “Dynamics and bifurcations of a planar map modeling dispersion managed breathers,” SIAM 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]

Kuznetsov, E. A.

E. A. Kuznetsov, A. V. Mikhailov, and I. A. Shimokhin, “Nonlinear interaction of solitons and radiation,” Physica D 87, 201–215 (1995).
[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]

Laing, E. W.

J. B. Taylor and E. W. Laing, “Invariant for a particle interacting with an electrostatic wave in a magnetic field,” Phys. Rev. Lett. 35, 1306–1307 (1975).
[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.

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]

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

Maruta, A.

McKinstrie, C. J.

Melnikov, L. A.

R. G. Bauer and L. A. Melnikov, “Multi-soliton fission and quasi-periodicity in a fiber with a periodically modulated core diameter,” Opt. Commun. 115, 190–198 (1995).
[CrossRef]

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]

Mikhailov, A. V.

E. A. Kuznetsov, A. V. Mikhailov, and I. A. Shimokhin, “Nonlinear interaction of solitons and radiation,” Physica D 87, 201–215 (1995).
[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 fiber 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]

Selaru, D.

F. Diacu and D. Şelaru, “Chaos in the Gyldén problem,” J. Math. Phys. 39, 6537–6546 (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]

Shimokhin, I. A.

E. A. Kuznetsov, A. V. Mikhailov, and I. A. Shimokhin, “Nonlinear interaction of solitons and radiation,” Physica D 87, 201–215 (1995).
[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 fiber repeater pumped by GaInAsP laser diode,” Electron. Lett. 25, 199–200 (1989).
[CrossRef]

Takashi, I.

Taylor, J. B.

J. B. Taylor and E. W. Laing, “Invariant for a particle interacting with an electrostatic wave in a magnetic field,” Phys. Rev. Lett. 35, 1306–1307 (1975).
[CrossRef]

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]

I. Gabitov and S. K. Turitsyn, “Averaged pulse dynamics in a cascaded transmission system with passive dispersion compensation,” Opt. Lett. 21, 327–329 (1996).
[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]

Zaslavsky, G. M.

G. M. Zaslavsky, “The width of exponentially narrow stochastic layers,” Chaos 4, 589–591 (1994).
[CrossRef] [PubMed]

Astron. Nachr. (1)

H. Gyldén, “Die Bahnbewegung in einem Systeme von zwei Körpern in dem Falle dass die Massen Veränderungen unterworfen sind,” Astron. Nachr. 109, 1–6 (1884).
[CrossRef]

Chaos (1)

G. M. Zaslavsky, “The width of exponentially narrow stochastic layers,” Chaos 4, 589–591 (1994).
[CrossRef] [PubMed]

Electron. Lett. (1)

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

J. Lightwave Technol. (1)

J. Math. Phys. (1)

F. Diacu and D. Şelaru, “Chaos in the Gyldén problem,” J. Math. Phys. 39, 6537–6546 (1998).
[CrossRef]

J. Opt. Soc. Am. B (7)

Opt. Commun. (5)

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]

R. G. Bauer and L. A. Melnikov, “Multi-soliton fission and quasi-periodicity in a fiber with a periodically modulated core diameter,” Opt. Commun. 115, 190–198 (1995).
[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]

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]

Opt. Lett. (1)

Phys. Rep. (1)

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

Phys. Rev. A (1)

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

Phys. Rev. E (3)

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]

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]

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]

Phys. Rev. Lett. (1)

J. B. Taylor and E. W. Laing, “Invariant for a particle interacting with an electrostatic wave in a magnetic field,” Phys. Rev. Lett. 35, 1306–1307 (1975).
[CrossRef]

Phys. Scr. (1)

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]

Physica D (1)

E. A. Kuznetsov, A. V. Mikhailov, and I. A. Shimokhin, “Nonlinear interaction of solitons and radiation,” Physica D 87, 201–215 (1995).
[CrossRef]

Quantum Electron. (1)

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]

SIAM J. Appl. Math. (1)

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

Other (4)

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

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

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

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

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Figures (10)

Fig. 1
Fig. 1

Propagation of a soliton of duration ts=4 ps under amplification period d=100 km. N=0.793, corresponding to ω0=Ω: (a) phase-plane diagram in terms of the original variables, (b) phase-plane diagram in terms of action–angle, (c) analytically constructed approximate invariant, and (d) the corresponding contour plot.

Fig. 2
Fig. 2

Propagation of a soliton of duration ts=4 ps under amplification period d=40 km. N=0.997, corresponding to ω0=Ω: (a) phase-plane diagram in terms of the original variables, (b) phase-plane diagram in terms of action–angle, (c) analytically constructed approximate invariant, and (d) the corresponding contour plot.

Fig. 3
Fig. 3

Propagation of a soliton of ts=8 ps duration under amplification period d=100 km. N=1.121, corresponding to ω0=Ω: (a) phase-plane diagram in terms of the original variables, (b) phase-plane diagram in terms of action–angle, (c) analytically constructed approximate invariant, and (d) the corresponding contour plot.

Fig. 4
Fig. 4

Propagation of a soliton of ts=8 ps duration under amplification period d=100 km. N=0.9, corresponding to ω0Ω: (a) phase-plane diagram in terms of the original variables, (b) phase-plane diagram in terms of action–angle, (c) analytically constructed approximate invariant, and (d) the corresponding contour plot.

Fig. 5
Fig. 5

Iterations of the separatrix map: (a) ts=4 ps duration under amplification period d=100 km, N=0.793, corresponding to ω0=Ω; (b) ts=4 ps duration under amplification period d=40 km, N=0.997, corresponding to ω0=Ω.

Fig. 6
Fig. 6

Nonlinear shape oscillations of a soliton of ts=8 ps duration under amplification period d=100 km. N=1.121, corresponding to ω0=Ω. The initial width and chirp correspond to the point (α, α)=(1.1, 0.1) in Fig. 3(a): (a) numerical integration of the NLS equation; (b) comparison of the direct simulation (solid curve) with the variational method (dashed curve).

Fig. 7
Fig. 7

Soliton spreading and decay of a soliton under the same configuration as in Fig. 6 but with initial width and chirp corresponding to a point (α, α)=(0.2, 0.5) (E0>0): (a) numerical integration of the NLS equation; (b) comparison of the direct simulation (solid curve) with the variational method (dashed curve).

Fig. 8
Fig. 8

Nonlinear shape oscillations of a soliton of ts=8 ps duration under amplification period d=100 km. N=0.9, corresponding to ω0Ω. The initial width and chirp correspond to the point (α, α)=(1.25,-0.04) in Fig. 4(a): (a) numerical integration of the NLS equation; (b) comparison of the direct simulation (solid curve) with the variational method (dashed curve).

Fig. 9
Fig. 9

Soliton spreading and decay of a soliton under the same configuration as in Fig. 8 but with initial width and chirp corresponding to a point (α, α)=(0.2, 0.1) (E0>0): (a) numerical integration of the NLS equation; (b) comparison of the direct simulation (solid curve) with the variational method (dashed curve).

Fig. 10
Fig. 10

Propagation and splitting of a soliton of ts=13.6 ps duration under amplification period d=40 km. N=1.83, corresponding to ω0=Ω. The initial width and chirp correspond to a point (α, α)=(1, 0.5): (a) numerical integration of the NLS equation; (b) comparison of the direct simulation (solid curve) with the variational method (dashed curve).

Equations (57)

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i qZ+12 2qT2+|q|2q=iγ0(Z)q.
Z0[m]=6.07×102 ts2[ps]λ2[μm]D[ps/nm·km],
T0=ts1.76,
γ0(Z)=-Γ0+Γ1n=-+δ(Z-Zan),
Γ0=7×10-2 ts2[ps]δ[dB/km]λ2[μm]D[ps/nm·km].
u(T, Z)=exp[-Γ(Z)]q(T, Z),
Γ(Z)=γ0(Z)dZ,
i uZ+12 2uT2+exp[2Γ(Z)]|u|2u=0.
u(Z, T)=A(Z)sechTα(Z)exp[iϕ0(Z)T2],
L=iu u*Z-u* uZ+uT2-exp[2Γ(Z)]|u|4
L=-LdT.
δ0TLdZ=0,
ϕ0=12 d(ln α)dZ,
d(α|A|2)dZ=0,α|A|2=N2=const.,
d2αdZ2=4π2α3-4N2 exp[Δ(Z)]π2α2,
d(arg A)dZ=-13α2+56 N2 exp[Δ(Z)]α,
Δ(Z)=4πΓ(Z)
Δ(Z)=4π2Γ1n0 n-12i expi 2πnZaZ.
U=2π2α2-4N2π2α,
E0=2π2α02-4N2π2α0+2α02ϕ02.
H0α, dαdZ=12 dαdZ2+2π2α2-4N2π2α=E.
α(Z)=mαm exp(-imωZ),αm=2bmJm(e0m),
α(Z)=b[1-e0 cos(ξ)],ωZ=ξ-e0 sin(ξ),
e0=1-π2|E|2N4,b=2N2π2|E|.
1α(Z)=mcm exp(-imωZ),cm=2bJm(e0m).
J=12π dαdZdZ=22N2π2-E-2π,
θ=ξ-e0 sin ξ.
H0(J)=-8N4π2 1(πJ+2)2=E,
ω(J)=dH0dJ=16N4π 1(πJ+2)3.
N0=18(πJ+2)3,
H(J, θ, Z)=H0(J)+H1(J, θ, Z),
H1(J, θ, Z)=-4N2π2 1α(J, θ)Δ(Z),
S=Jθ¯+S(J¯, θ, Z)
H¯=H(J, θ, Z)+ S1(J¯, θ, Z)Z.
H¯0=H0(J¯),
H¯1=S1Z+ω S1θ¯+H1.
H¯=H0+H1,
S1Z+ω S1θ¯=-{H1}.
H1=0,
{H1}=16N2π4 Γ1b mn0in-1Jm(e0m)exp(inΩZ-imθ),
Δω=H1J=0.
S1=-16N2π4 Γ1b mn0 n-1Jm(e0m)nΩ-mω(J) exp(inΩZ-imθ).
J¯=J- 16N2π4 Γ1b m,n0 imn-1Jm(e0m)nΩ-mω(J)×exp(inΩZ-imθ)=const.
mω(Jmn)-nΩ=0
I=I0(J)- dI0dJ 16N2π4 Γ1b ×m,n0 imn-1Jm(e0m)nΩ-mω(J) exp(inΩZ-imθ)=const.
dI0dJ=sinπΩω(J),
ΔJmn=22 16N2π4 Γ1bn-1Jm(e0m)dωdJJ=Jmn1/2.
s=12 ΔJmn+ΔJmnδJmn,
M(Z0)=-+[H0, H1][α0(Z)]dZ,
[H0, H1]=H0α H1αz-H0αz H1α,
M(Z0)=-4N2π2 -+ α0z(Z)α02(Z)Δ(Z+Z0)dZ=-16N2π4Γ1n=1n-1 cos(nΩZ0)In,
In=-+ α0z(Z)α02(Z) sin(nΩZ)dZ.
α0(Z)=11+cos v,
Z=k34 tanv2+13 tan3v2,
In=43Ωk3 sinπ3K1/3n2N2Ωk33π2,
Ek+1=Ek+ΔEϕkΩ,
ϕk+1=ϕk+ΩT(Ek+1).

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