Abstract

We use the variational method to find the optimum launch conditions that can sustain path-averaged solitons in a periodically amplified, constant-dispersion, optical communication system even when amplifier spacing is comparable to or larger than the dispersion length. We determine the amount of prechirping and the initial peak power required and show that both the pulse width and the chirp recover their initial values at each amplifier. The prechirped solitons are different from the standard solitons in constant dispersion since their width and chirp are allowed to vary over each amplifier section. This feature results in an interesting regime in which amplifier spacing can exceed the dispersion length. Numerical solutions of the nonlinear Schrödinger equation show that the use of prechirped solitons improves stability in comparison with guiding-center solitons in constant-dispersion fiber links.

© 2000 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. P. Agrawal, Fiber-Optic Communication Systems, 2nd ed. (Wiley, New York, 1997).
  2. A. Hasegawa and Y. Kodama, Solitons in Optical Communications (Clarendon, Oxford, UK, 1995).
  3. A. Hasegawa and Y. Kodama, “Guiding-center soliton in optical fibers,” Opt. Lett. 15, 1443–1445 (1990); “Guiding-center soliton,” Phys. Rev. Lett. 66, 161–164 (1991).
    [CrossRef] [PubMed]
  4. L. F. Mollenauer, S. G. Evangelides, and H. A. Haus, “Long-distance soliton propagation using lumped amplifiers and dispersion shifted fiber,” J. Lightwave Technol. 9, 194–196 (1991).
    [CrossRef]
  5. K. J. Blow and N. J. Doran, “Average soliton dynamics and the operation of soliton systems with lumped amplifiers,” IEEE Photonics Technol. Lett. 3, 369–371 (1991).
    [CrossRef]
  6. S. M. J. Kelly, “Characteristic sideband instability of periodically amplified average soliton,” Electron. Lett. 28, 806–807 (1992).
    [CrossRef]
  7. N. J. Smith, K. J. Blow, and I. Andonovic, “Sideband generation through perturbations to the average soliton model,” J. Lightwave Technol. 10, 1329–1333 (1992).
    [CrossRef]
  8. J. P. Gordan, “Dispersive perturbations of solitons of the nonlinear Schrödinger equation,” J. Opt. Soc. Am. B 9, 91–97 (1992).
    [CrossRef]
  9. 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]
  10. M. Matsumoto, H. Ikeda, T. Uda, and A. Hasegawa, “Stable soliton transmission in the system with nonlinear gain,” J. Lightwave Technol. 13, 658–665 (1995).
    [CrossRef]
  11. 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]
  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. Z. M. Liao and G. P. Agrawal, “High-bit-rate soliton transmission using distributed amplification and dispersion management,” IEEE Photonics Technol. Lett. 11, 818–820 (1999).
    [CrossRef]
  14. T. Okamawari, Y. Ueda, A. Maruta, Y. Kodama, and A. Hasegawa, “Interaction between guiding centre solitons in a periodically compensated optical transmission line,” Electron. Lett. 33, 1063–1065 (1997).
    [CrossRef]
  15. Y. Kodama and A. Maruta, “Optimal design of dispersion management for a soliton-wavelength-division-multiplexed system,” Opt. Lett. 22, 1692–1694 (1997).
    [CrossRef]
  16. I. Morita, M. Suzuki, N. Edagawa, K. Tanaka, S. Yamamoto, and S. Akiba, “Performance improvement by initial phase modulation in 20 Gbit/s soliton-based RZ transmission with periodic dispersion compensation,” Electron. Lett. 33, 1021–1022 (1997).
    [CrossRef]
  17. H. Sugahara, T. Inoue, A. Maruta, and Y. Kodama, “Optimal dispersion management for wavelength-division-multiplexed RZ optical pulse transmission,” Electron. Lett. 34, 902–904 (1998).
    [CrossRef]
  18. T. Georges and B. Charbonnier, “Reduction of the dispersive wave in periodically amplified links with initially chirped solitons,” IEEE Photonics Technol. Lett. 9, 127–129 (1997).
    [CrossRef]
  19. F. Favre, D. Guen, and T. Georges, “Experimental evidence of pseudoperiodical soliton propagation in dispersion-managed links,” J. Lightwave Technol. 17, 1032–1036 (1999).
    [CrossRef]
  20. D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 27, 3135–3145 (1983).
    [CrossRef]
  21. W. L. Kath and N. F. Smyth, “Soliton evolution and radiation loss for the nonlinear Schrödinger equation,” Phys. Rev. E 51, 1484–1492 (1995).
    [CrossRef]

1999 (2)

Z. M. Liao and G. P. Agrawal, “High-bit-rate soliton transmission using distributed amplification and dispersion management,” IEEE Photonics Technol. Lett. 11, 818–820 (1999).
[CrossRef]

F. Favre, D. Guen, and T. Georges, “Experimental evidence of pseudoperiodical soliton propagation in dispersion-managed links,” J. Lightwave Technol. 17, 1032–1036 (1999).
[CrossRef]

1998 (1)

H. Sugahara, T. Inoue, A. Maruta, and Y. Kodama, “Optimal dispersion management for wavelength-division-multiplexed RZ optical pulse transmission,” Electron. Lett. 34, 902–904 (1998).
[CrossRef]

1997 (4)

T. Georges and B. Charbonnier, “Reduction of the dispersive wave in periodically amplified links with initially chirped solitons,” IEEE Photonics Technol. Lett. 9, 127–129 (1997).
[CrossRef]

Y. Kodama and A. Maruta, “Optimal design of dispersion management for a soliton-wavelength-division-multiplexed system,” Opt. Lett. 22, 1692–1694 (1997).
[CrossRef]

T. Okamawari, Y. Ueda, A. Maruta, Y. Kodama, and A. Hasegawa, “Interaction between guiding centre solitons in a periodically compensated optical transmission line,” Electron. Lett. 33, 1063–1065 (1997).
[CrossRef]

I. Morita, M. Suzuki, N. Edagawa, K. Tanaka, S. Yamamoto, and S. Akiba, “Performance improvement by initial phase modulation in 20 Gbit/s soliton-based RZ transmission with periodic dispersion compensation,” Electron. Lett. 33, 1021–1022 (1997).
[CrossRef]

1995 (5)

M. Matsumoto, H. Ikeda, T. Uda, and A. Hasegawa, “Stable soliton transmission in the system with nonlinear gain,” J. Lightwave Technol. 13, 658–665 (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]

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]

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

W. L. Kath and N. F. Smyth, “Soliton evolution and radiation loss for the nonlinear Schrödinger equation,” Phys. Rev. E 51, 1484–1492 (1995).
[CrossRef]

1992 (3)

J. P. Gordan, “Dispersive perturbations of solitons of the nonlinear Schrödinger equation,” J. Opt. Soc. Am. B 9, 91–97 (1992).
[CrossRef]

S. M. J. Kelly, “Characteristic sideband instability of periodically amplified average soliton,” Electron. Lett. 28, 806–807 (1992).
[CrossRef]

N. J. Smith, K. J. Blow, and I. Andonovic, “Sideband generation through perturbations to the average soliton model,” J. Lightwave Technol. 10, 1329–1333 (1992).
[CrossRef]

1991 (2)

L. F. Mollenauer, S. G. Evangelides, and H. A. Haus, “Long-distance soliton propagation using lumped amplifiers and dispersion shifted fiber,” J. Lightwave Technol. 9, 194–196 (1991).
[CrossRef]

K. J. Blow and N. J. Doran, “Average soliton dynamics and the operation of soliton systems with lumped amplifiers,” IEEE Photonics Technol. Lett. 3, 369–371 (1991).
[CrossRef]

1983 (1)

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

Agrawal, G. P.

Z. M. Liao and G. P. Agrawal, “High-bit-rate soliton transmission using distributed amplification and dispersion management,” IEEE Photonics Technol. Lett. 11, 818–820 (1999).
[CrossRef]

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

Akiba, S.

I. Morita, M. Suzuki, N. Edagawa, K. Tanaka, S. Yamamoto, and S. Akiba, “Performance improvement by initial phase modulation in 20 Gbit/s soliton-based RZ transmission with periodic dispersion compensation,” Electron. Lett. 33, 1021–1022 (1997).
[CrossRef]

Anderson, D.

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

Andonovic, I.

N. J. Smith, K. J. Blow, and I. Andonovic, “Sideband generation through perturbations to the average soliton model,” J. Lightwave Technol. 10, 1329–1333 (1992).
[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]

N. J. Smith, K. J. Blow, and I. Andonovic, “Sideband generation through perturbations to the average soliton model,” J. Lightwave Technol. 10, 1329–1333 (1992).
[CrossRef]

K. J. Blow and N. J. Doran, “Average soliton dynamics and the operation of soliton systems with lumped amplifiers,” IEEE Photonics Technol. Lett. 3, 369–371 (1991).
[CrossRef]

Charbonnier, B.

T. Georges and B. Charbonnier, “Reduction of the dispersive wave in periodically amplified links with initially chirped solitons,” IEEE Photonics Technol. Lett. 9, 127–129 (1997).
[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 transmission using concatenated nonlinear optical loop-mirror intensity filters,” J. Opt. Soc. Am. B 12, 1117–1125 (1995).
[CrossRef]

K. J. Blow and N. J. Doran, “Average soliton dynamics and the operation of soliton systems with lumped amplifiers,” IEEE Photonics Technol. Lett. 3, 369–371 (1991).
[CrossRef]

Edagawa, N.

I. Morita, M. Suzuki, N. Edagawa, K. Tanaka, S. Yamamoto, and S. Akiba, “Performance improvement by initial phase modulation in 20 Gbit/s soliton-based RZ transmission with periodic dispersion compensation,” Electron. Lett. 33, 1021–1022 (1997).
[CrossRef]

Essiambre, R. J.

Evangelides, S. G.

L. F. Mollenauer, S. G. Evangelides, and H. A. Haus, “Long-distance soliton propagation using lumped amplifiers and dispersion shifted fiber,” J. Lightwave Technol. 9, 194–196 (1991).
[CrossRef]

Favre, F.

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]

Georges, T.

F. Favre, D. Guen, and T. Georges, “Experimental evidence of pseudoperiodical soliton propagation in dispersion-managed links,” J. Lightwave Technol. 17, 1032–1036 (1999).
[CrossRef]

T. Georges and B. Charbonnier, “Reduction of the dispersive wave in periodically amplified links with initially chirped solitons,” IEEE Photonics Technol. Lett. 9, 127–129 (1997).
[CrossRef]

Gordan, J. P.

Guen, D.

Hasegawa, A.

T. Okamawari, Y. Ueda, A. Maruta, Y. Kodama, and A. Hasegawa, “Interaction between guiding centre solitons in a periodically compensated optical transmission line,” Electron. Lett. 33, 1063–1065 (1997).
[CrossRef]

M. Matsumoto, H. Ikeda, T. Uda, and A. Hasegawa, “Stable soliton transmission in the system with nonlinear gain,” J. Lightwave Technol. 13, 658–665 (1995).
[CrossRef]

Haus, H. A.

L. F. Mollenauer, S. G. Evangelides, and H. A. Haus, “Long-distance soliton propagation using lumped amplifiers and dispersion shifted fiber,” J. Lightwave Technol. 9, 194–196 (1991).
[CrossRef]

Ikeda, H.

M. Matsumoto, H. Ikeda, T. Uda, and A. Hasegawa, “Stable soliton transmission in the system with nonlinear gain,” J. Lightwave Technol. 13, 658–665 (1995).
[CrossRef]

Inoue, T.

H. Sugahara, T. Inoue, A. Maruta, and Y. Kodama, “Optimal dispersion management for wavelength-division-multiplexed RZ optical pulse transmission,” Electron. Lett. 34, 902–904 (1998).
[CrossRef]

Kath, W. L.

W. L. Kath and N. F. Smyth, “Soliton evolution and radiation loss for the nonlinear Schrödinger equation,” Phys. Rev. E 51, 1484–1492 (1995).
[CrossRef]

Kelly, S. M. J.

S. M. J. Kelly, “Characteristic sideband instability of periodically amplified average soliton,” Electron. Lett. 28, 806–807 (1992).
[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.

H. Sugahara, T. Inoue, A. Maruta, and Y. Kodama, “Optimal dispersion management for wavelength-division-multiplexed RZ optical pulse transmission,” Electron. Lett. 34, 902–904 (1998).
[CrossRef]

Y. Kodama and A. Maruta, “Optimal design of dispersion management for a soliton-wavelength-division-multiplexed system,” Opt. Lett. 22, 1692–1694 (1997).
[CrossRef]

T. Okamawari, Y. Ueda, A. Maruta, Y. Kodama, and A. Hasegawa, “Interaction between guiding centre solitons in a periodically compensated optical transmission line,” Electron. Lett. 33, 1063–1065 (1997).
[CrossRef]

Liao, Z. M.

Z. M. Liao and G. P. Agrawal, “High-bit-rate soliton transmission using distributed amplification and dispersion management,” IEEE Photonics Technol. Lett. 11, 818–820 (1999).
[CrossRef]

Maruta, A.

H. Sugahara, T. Inoue, A. Maruta, and Y. Kodama, “Optimal dispersion management for wavelength-division-multiplexed RZ optical pulse transmission,” Electron. Lett. 34, 902–904 (1998).
[CrossRef]

Y. Kodama and A. Maruta, “Optimal design of dispersion management for a soliton-wavelength-division-multiplexed system,” Opt. Lett. 22, 1692–1694 (1997).
[CrossRef]

T. Okamawari, Y. Ueda, A. Maruta, Y. Kodama, and A. Hasegawa, “Interaction between guiding centre solitons in a periodically compensated optical transmission line,” Electron. Lett. 33, 1063–1065 (1997).
[CrossRef]

Matsumoto, M.

M. Matsumoto, H. Ikeda, T. Uda, and A. Hasegawa, “Stable soliton transmission in the system with nonlinear gain,” J. Lightwave Technol. 13, 658–665 (1995).
[CrossRef]

Mollenauer, L. F.

L. F. Mollenauer, S. G. Evangelides, and H. A. Haus, “Long-distance soliton propagation using lumped amplifiers and dispersion shifted fiber,” J. Lightwave Technol. 9, 194–196 (1991).
[CrossRef]

Morita, I.

I. Morita, M. Suzuki, N. Edagawa, K. Tanaka, S. Yamamoto, and S. Akiba, “Performance improvement by initial phase modulation in 20 Gbit/s soliton-based RZ transmission with periodic dispersion compensation,” Electron. Lett. 33, 1021–1022 (1997).
[CrossRef]

Okamawari, T.

T. Okamawari, Y. Ueda, A. Maruta, Y. Kodama, and A. Hasegawa, “Interaction between guiding centre solitons in a periodically compensated optical transmission line,” Electron. Lett. 33, 1063–1065 (1997).
[CrossRef]

Smith, 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]

N. J. Smith, K. J. Blow, and I. Andonovic, “Sideband generation through perturbations to the average soliton model,” J. Lightwave Technol. 10, 1329–1333 (1992).
[CrossRef]

Smyth, N. F.

W. L. Kath and N. F. Smyth, “Soliton evolution and radiation loss for the nonlinear Schrödinger equation,” Phys. Rev. E 51, 1484–1492 (1995).
[CrossRef]

Sugahara, H.

H. Sugahara, T. Inoue, A. Maruta, and Y. Kodama, “Optimal dispersion management for wavelength-division-multiplexed RZ optical pulse transmission,” Electron. Lett. 34, 902–904 (1998).
[CrossRef]

Suzuki, M.

I. Morita, M. Suzuki, N. Edagawa, K. Tanaka, S. Yamamoto, and S. Akiba, “Performance improvement by initial phase modulation in 20 Gbit/s soliton-based RZ transmission with periodic dispersion compensation,” Electron. Lett. 33, 1021–1022 (1997).
[CrossRef]

Tanaka, K.

I. Morita, M. Suzuki, N. Edagawa, K. Tanaka, S. Yamamoto, and S. Akiba, “Performance improvement by initial phase modulation in 20 Gbit/s soliton-based RZ transmission with periodic dispersion compensation,” Electron. Lett. 33, 1021–1022 (1997).
[CrossRef]

Uda, T.

M. Matsumoto, H. Ikeda, T. Uda, and A. Hasegawa, “Stable soliton transmission in the system with nonlinear gain,” J. Lightwave Technol. 13, 658–665 (1995).
[CrossRef]

Ueda, Y.

T. Okamawari, Y. Ueda, A. Maruta, Y. Kodama, and A. Hasegawa, “Interaction between guiding centre solitons in a periodically compensated optical transmission line,” Electron. Lett. 33, 1063–1065 (1997).
[CrossRef]

Yamamoto, S.

I. Morita, M. Suzuki, N. Edagawa, K. Tanaka, S. Yamamoto, and S. Akiba, “Performance improvement by initial phase modulation in 20 Gbit/s soliton-based RZ transmission with periodic dispersion compensation,” Electron. Lett. 33, 1021–1022 (1997).
[CrossRef]

Electron. Lett. (4)

S. M. J. Kelly, “Characteristic sideband instability of periodically amplified average soliton,” Electron. Lett. 28, 806–807 (1992).
[CrossRef]

I. Morita, M. Suzuki, N. Edagawa, K. Tanaka, S. Yamamoto, and S. Akiba, “Performance improvement by initial phase modulation in 20 Gbit/s soliton-based RZ transmission with periodic dispersion compensation,” Electron. Lett. 33, 1021–1022 (1997).
[CrossRef]

H. Sugahara, T. Inoue, A. Maruta, and Y. Kodama, “Optimal dispersion management for wavelength-division-multiplexed RZ optical pulse transmission,” Electron. Lett. 34, 902–904 (1998).
[CrossRef]

T. Okamawari, Y. Ueda, A. Maruta, Y. Kodama, and A. Hasegawa, “Interaction between guiding centre solitons in a periodically compensated optical transmission line,” Electron. Lett. 33, 1063–1065 (1997).
[CrossRef]

IEEE Photonics Technol. Lett. (3)

T. Georges and B. Charbonnier, “Reduction of the dispersive wave in periodically amplified links with initially chirped solitons,” IEEE Photonics Technol. Lett. 9, 127–129 (1997).
[CrossRef]

Z. M. Liao and G. P. Agrawal, “High-bit-rate soliton transmission using distributed amplification and dispersion management,” IEEE Photonics Technol. Lett. 11, 818–820 (1999).
[CrossRef]

K. J. Blow and N. J. Doran, “Average soliton dynamics and the operation of soliton systems with lumped amplifiers,” IEEE Photonics Technol. Lett. 3, 369–371 (1991).
[CrossRef]

J. Lightwave Technol. (4)

L. F. Mollenauer, S. G. Evangelides, and H. A. Haus, “Long-distance soliton propagation using lumped amplifiers and dispersion shifted fiber,” J. Lightwave Technol. 9, 194–196 (1991).
[CrossRef]

N. J. Smith, K. J. Blow, and I. Andonovic, “Sideband generation through perturbations to the average soliton model,” J. Lightwave Technol. 10, 1329–1333 (1992).
[CrossRef]

M. Matsumoto, H. Ikeda, T. Uda, and A. Hasegawa, “Stable soliton transmission in the system with nonlinear gain,” J. Lightwave Technol. 13, 658–665 (1995).
[CrossRef]

F. Favre, D. Guen, and T. Georges, “Experimental evidence of pseudoperiodical soliton propagation in dispersion-managed links,” J. Lightwave Technol. 17, 1032–1036 (1999).
[CrossRef]

J. Opt. Soc. Am. B (3)

Opt. Commun. (1)

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]

Opt. Lett. (1)

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 (1)

W. L. Kath and N. F. Smyth, “Soliton evolution and radiation loss for the nonlinear Schrödinger equation,” Phys. Rev. E 51, 1484–1492 (1995).
[CrossRef]

Other (3)

G. P. Agrawal, Fiber-Optic Communication Systems, 2nd ed. (Wiley, New York, 1997).

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

A. Hasegawa and Y. Kodama, “Guiding-center soliton in optical fibers,” Opt. Lett. 15, 1443–1445 (1990); “Guiding-center soliton,” Phys. Rev. Lett. 66, 161–164 (1991).
[CrossRef] [PubMed]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

Comparison of (a) launching peak power and (b) initial chirp for chirped (solid curves) and unchirped (dotted curves) solitons as a function of normalized amplifier spacing when amplifiers with 10-dB gain (Γ=2.3) are placed 40 km apart.

Fig. 2
Fig. 2

Evolution of pulse width and chirp over one amplifier stage for chirped (solid curves) and unchirped (dotted curves) solitons as predicted by variational analysis. Normalized amplifier spacing zA=0.4 for top row and 2.1 for bottom row.

Fig. 3
Fig. 3

Same as in Fig. 2 except that soliton evolution over 20 amplification stages (total distance of 800 km) is shown by solving the NSE numerically.

Fig. 4
Fig. 4

Poincaré map obtained by plotting soliton width and chirp at the end of each amplifier section for 100 amplification stages (4000 km) for unchirped (left graph) and chirped (right graph) solitons. For zA=0.4, a nearly circular compact region shows the quasi-periodic nature of soliton evolution. For zA=2.1, soliton width and chirp vary over a wider region.

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

i Az-12β2 2At2+γ0|A|2A=-i2αA,
A=B exp(-αz/2),
i Bz-12β2 2Bt2+γ(z)|B|2B=0,
B(z,t)=a sechtτexpiϕ-iCt22τ2,
ddz(a2τ)=0,
dτdz=β2Cτ,
dCdz=4π2γ(z)a2+β2τ24π2+C2,
dϕdz=β23τ2+56γ(z)a2.
dWdξ=-zACW,
dCdξ=4zAP0 exp(-Γξ)π2W-zAW24π2+C2,
C(0)=C(1),W(0)=W(1)1.
W=W0+W1zA+W2zA2,
C=C0+C1zA+C2zA2.
dW2dξ=-C1,
dC1dξ=4P0 exp(-Γξ)π2-4π2.
P0=Γ1-exp(-Γ)=G ln GG-1,
C1(0)=2π2-4π2 exp(-Γ)+Γ-1Γ[1-exp(-Γ)]=4π212+(G-1)-G ln Gln G(G-1).
01C(ξ)dξ=0.

Metrics