Abstract

We analytically describe the effective evolution of a pulse (nonreturn-to-zero or return-to-zero) that propagates under the influence of a mean-zero dispersion map, nonlinearity, loss, and periodic amplification. On averaging, the governing equation is reduced to a set of coupled, nonlinear diffusion equations that describe the evolution of the pulse amplitude and phase and which capture the long-term interaction of dispersion and nonlinearity. The averaged equations are shown to be in good agreement with the full evolution until a predicted wave-breaking behavior is observed in the full equations.

© 1997 Optical Society of America

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  1. N. S. Bergano, C. R. Davidson, D. L. Wilson, F. W. Kerfoot, M. D. Tremblay, M. D. Levonas, J. P. Morreale, J. D. Evankow, P. C. Corbett, M. A. Mills, G. A. Ferguson, A. M. Vengsarkar, J. R. Predrazzani, J. A. Nagel, J. L. Zyskind, and J. W. Sulhoff, “100 Gb/s error free transmission over 9,100 km using twenty 5 Gb/s WDM data channels,” in Optical Fiber Communication Conference Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD23.
  2. N. S. Bergano and C. R. Davidson, “40 Gb/s WDM transmission of eight 5 Gb/s data channels over transoceanic distances using the conventional NRZ modulation format,” in Optical Fiber Communication Conference, Vol. 8 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), paper PD19.
  3. A. H. Gnauck, A. R. Chraplyvy, R. W. Tkach, and R. M. Derosier, “160 Gbit/s (8×20 Gbit/s WDM) 300 km transmission with 50 km amplifier spacing and span-by-span dispersion reversal,” Electron. Lett. 30, 1241–1243 (1994).
    [CrossRef]
  4. A. R. Chraplyvy, A. H. Gnauck, R. W. Tkach, R. M. Derosier, C. R. Giles, B. M. Nyman, G. A. Ferguson, J. W. Sulhoff, and J. L. Zyskind, “1/3rd-Terabit/s transmission through 150-km of dispersion-managed fiber,” IEEE Photonics Technol. Lett. 7, 98–100 (1995).
    [CrossRef]
  5. A. H. Gnauck, A. R. Chraplyvy, R. W. Tkach, J. L. Zyskind, J. W. Sulhoff, A. J. Lucero, Y. Sun, R. M. Jopson, F. Forguieri, R. M. Derosier, C. Wolf, and A. R. McCormick, “One terabit/s transmission experiment,” in Optical Fiber Communication Conference, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD20.
  6. H. Onaka, H. Miyata, G. Ishikawa, K. Otsuka, H. Ooi, Y. Kai, S. Kinoshita, M. Seino, H. Nishimito, and T. Chikama, “1.1 Tb/s WDM transmission over a 150 km 1.3 µm zero-dispersion single-mode fiber,” in Optical Fiber Communication Conference, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD19.
  7. R. W. Tkach, A. R. Chraplyvy, F. Forguieri, A. H. Gnauck, and R. M. Derosier “4-Photon mixing and high-speed WDM systems,” J. Lightwave Technol. 13, 841–849 (1995).
    [CrossRef]
  8. C. Kurtzke, “Suppression of fiber nonlinearities by appropriate dispersion management,” IEEE Photonics Technol. Lett. 5, 1250–1253 (1993).
    [CrossRef]
  9. L. F. Mollenauer, E. Lichtman, G. T. Harvey, M. J. Neubelt, and B. M. Nyman, “Demonstration of error-free soliton transmission over more than 15000 km at 2.5 Gbit/s, single-channel, and over more than 11000 km at 10 Gbit/s in 2-channel WDM,” Electron. Lett. 28, 792–794 (1992).
    [CrossRef]
  10. M. Nakazawa, K. Susuki, E. Yamada, H. Kubota, Y. Kimura, and M. Takaya, “Experimental demonstration of soliton data-transmission over unlimited distances with soliton control in time and frequency domains,” Electron. Lett. 29, 729–730 (1993).
    [CrossRef]
  11. J. C. Bronski and J. N. Kutz, “Modulational stability of plane waves in nonreturn-to-zero communications systems with dispersion management,” Opt. Lett. 21, 937–939 (1996).
    [CrossRef] [PubMed]
  12. J. C. Bronski and J. N. Kutz, “Asymptotic behavior of the nonlinear Schrödinger equation with rapidly varying, mean-zero dispersion,” Physica D (to be published).
  13. Y. Kodama and S. Wabnitz, “Analytical theory of guiding-center nonreturn-to-zero and return-to-zero signal transmission in normally dispersive nonlinear optical fibers,” Opt. Lett. 20, 2291–2293 (1995).
    [CrossRef] [PubMed]
  14. I. R. Gabitov and S. K. Turitsyn, “Averaged pulse dynamics in a cascaded transmission system with passive dispersion compensation,” Opt. Lett. 21, 327–329 (1996).
    [CrossRef]
  15. K. H. Spatshek, S. K. Turitsyn, and Y. S. Kivshar, “Average envelope soliton dynamics in systems with periodically varying dispersion,” Phys. Lett. A 204, 269–273 (1995).
    [CrossRef]
  16. A. Hasegawa and Y. Kodama, “Guiding-center soliton in optical fibers,” Opt. Lett. 15, 1443–1445 (1990).
    [CrossRef] [PubMed]
  17. A. Hasegawa and Y. Kodama, “Guiding-center soliton in fibers with periodically varying dispersion,” Opt. Lett. 16, 1385–1387 (1991).
    [CrossRef] [PubMed]
  18. A. M. Weiner, W. J. Tomlinson, R. N. Thurston, D. E. Leaird, J. P. Heritage, E. M. Kirschner, and R. J. Hawkins, “Experimental-observation of the fundamental dark soliton in optical fibers,” Phys. Rev. Lett. 61, 2445–2448 (1988).
    [CrossRef] [PubMed]
  19. A. Hasegawa and Y. Kodama, “Guiding-center soliton,” Phys. Rev. Lett. 66, 161–164 (1991).
    [CrossRef] [PubMed]
  20. L. F. Mollenauer, S. G. Evangelides, and J. P. Gordon, “Wavelength division multiplexing with solitons in ultra-long distance transmission using lumped amplifiers,” J. Lightwave Technol. 9, 362–367 (1991).
    [CrossRef]
  21. G. I. Barenblatt, Similarity, Self-similarity, and Intermediate Asymptotics (Consultants Bureau, New York, 1979).
  22. E. Zauderer, Partial Differential Equations of Applied Mathematics (Wiley, 1989), Chap. 3, pp. 136–139.
  23. T. Y. Hou, J. S. Lowengrub, and M. J. Shelley, “Removing the stiffness from interfacial flows with surface tension,” J. Comput. Phys. 114, 312–338 (1994).
    [CrossRef]
  24. G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1989), Chap. 2, pp. 44–48.
  25. T. R. Taha and M. J. Ablowitz, “Analytical and numerical aspects of certain nonlinear evolution equations: II. Numerical, nonlinear Schrödinger equation,” J. Comput. Phys. 55, 203–230 (1984).
    [CrossRef]
  26. J. E. Rothenberg and D. Grishkowsky, “Observation of the formation of an optical-intensity shock and wave breaking in the nonlinear propagation of pulses in optical fibers,” Phys. Rev. Lett. 62, 531–534 (1989).
    [CrossRef] [PubMed]
  27. N. Tzoar and M. Jain, “Self-phase modulation in long-geometry optical waveguides,” Phys. Rev. A 23, 1266–1270 (1981).
    [CrossRef]
  28. J. E. Rothenberg, “Femtosecond optical shocks and wave breaking in fiber propagation,” J. Opt. Soc. Am. B 6, 2392–2401 (1989).
    [CrossRef]
  29. J. C. Bronski and D. W. McLaughlin, “Semiclassical behavior in the NLS equation: optical shocks—focusing instabilities,” in Singular Limits of Dispersive Waves, N. M. Ercolani, ed. (Plenum, New York, 1994), pp. 21–38.
  30. Y. Kodama, “Normal forms for weakly dispersive wave equations,” Phys. Lett. 112A, 193–196 (1985).
    [CrossRef]
  31. C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978), Chap. 7.

1996 (2)

1995 (4)

K. H. Spatshek, S. K. Turitsyn, and Y. S. Kivshar, “Average envelope soliton dynamics in systems with periodically varying dispersion,” Phys. Lett. A 204, 269–273 (1995).
[CrossRef]

Y. Kodama and S. Wabnitz, “Analytical theory of guiding-center nonreturn-to-zero and return-to-zero signal transmission in normally dispersive nonlinear optical fibers,” Opt. Lett. 20, 2291–2293 (1995).
[CrossRef] [PubMed]

A. R. Chraplyvy, A. H. Gnauck, R. W. Tkach, R. M. Derosier, C. R. Giles, B. M. Nyman, G. A. Ferguson, J. W. Sulhoff, and J. L. Zyskind, “1/3rd-Terabit/s transmission through 150-km of dispersion-managed fiber,” IEEE Photonics Technol. Lett. 7, 98–100 (1995).
[CrossRef]

R. W. Tkach, A. R. Chraplyvy, F. Forguieri, A. H. Gnauck, and R. M. Derosier “4-Photon mixing and high-speed WDM systems,” J. Lightwave Technol. 13, 841–849 (1995).
[CrossRef]

1994 (2)

A. H. Gnauck, A. R. Chraplyvy, R. W. Tkach, and R. M. Derosier, “160 Gbit/s (8×20 Gbit/s WDM) 300 km transmission with 50 km amplifier spacing and span-by-span dispersion reversal,” Electron. Lett. 30, 1241–1243 (1994).
[CrossRef]

T. Y. Hou, J. S. Lowengrub, and M. J. Shelley, “Removing the stiffness from interfacial flows with surface tension,” J. Comput. Phys. 114, 312–338 (1994).
[CrossRef]

1993 (2)

M. Nakazawa, K. Susuki, E. Yamada, H. Kubota, Y. Kimura, and M. Takaya, “Experimental demonstration of soliton data-transmission over unlimited distances with soliton control in time and frequency domains,” Electron. Lett. 29, 729–730 (1993).
[CrossRef]

C. Kurtzke, “Suppression of fiber nonlinearities by appropriate dispersion management,” IEEE Photonics Technol. Lett. 5, 1250–1253 (1993).
[CrossRef]

1992 (1)

L. F. Mollenauer, E. Lichtman, G. T. Harvey, M. J. Neubelt, and B. M. Nyman, “Demonstration of error-free soliton transmission over more than 15000 km at 2.5 Gbit/s, single-channel, and over more than 11000 km at 10 Gbit/s in 2-channel WDM,” Electron. Lett. 28, 792–794 (1992).
[CrossRef]

1991 (3)

A. Hasegawa and Y. Kodama, “Guiding-center soliton,” Phys. Rev. Lett. 66, 161–164 (1991).
[CrossRef] [PubMed]

L. F. Mollenauer, S. G. Evangelides, and J. P. Gordon, “Wavelength division multiplexing with solitons in ultra-long distance transmission using lumped amplifiers,” J. Lightwave Technol. 9, 362–367 (1991).
[CrossRef]

A. Hasegawa and Y. Kodama, “Guiding-center soliton in fibers with periodically varying dispersion,” Opt. Lett. 16, 1385–1387 (1991).
[CrossRef] [PubMed]

1990 (1)

1989 (2)

J. E. Rothenberg and D. Grishkowsky, “Observation of the formation of an optical-intensity shock and wave breaking in the nonlinear propagation of pulses in optical fibers,” Phys. Rev. Lett. 62, 531–534 (1989).
[CrossRef] [PubMed]

J. E. Rothenberg, “Femtosecond optical shocks and wave breaking in fiber propagation,” J. Opt. Soc. Am. B 6, 2392–2401 (1989).
[CrossRef]

1988 (1)

A. M. Weiner, W. J. Tomlinson, R. N. Thurston, D. E. Leaird, J. P. Heritage, E. M. Kirschner, and R. J. Hawkins, “Experimental-observation of the fundamental dark soliton in optical fibers,” Phys. Rev. Lett. 61, 2445–2448 (1988).
[CrossRef] [PubMed]

1985 (1)

Y. Kodama, “Normal forms for weakly dispersive wave equations,” Phys. Lett. 112A, 193–196 (1985).
[CrossRef]

1984 (1)

T. R. Taha and M. J. Ablowitz, “Analytical and numerical aspects of certain nonlinear evolution equations: II. Numerical, nonlinear Schrödinger equation,” J. Comput. Phys. 55, 203–230 (1984).
[CrossRef]

1981 (1)

N. Tzoar and M. Jain, “Self-phase modulation in long-geometry optical waveguides,” Phys. Rev. A 23, 1266–1270 (1981).
[CrossRef]

Ablowitz, M. J.

T. R. Taha and M. J. Ablowitz, “Analytical and numerical aspects of certain nonlinear evolution equations: II. Numerical, nonlinear Schrödinger equation,” J. Comput. Phys. 55, 203–230 (1984).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1989), Chap. 2, pp. 44–48.

Barenblatt, G. I.

G. I. Barenblatt, Similarity, Self-similarity, and Intermediate Asymptotics (Consultants Bureau, New York, 1979).

Bender, C. M.

C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978), Chap. 7.

Bergano, N. S.

N. S. Bergano, C. R. Davidson, D. L. Wilson, F. W. Kerfoot, M. D. Tremblay, M. D. Levonas, J. P. Morreale, J. D. Evankow, P. C. Corbett, M. A. Mills, G. A. Ferguson, A. M. Vengsarkar, J. R. Predrazzani, J. A. Nagel, J. L. Zyskind, and J. W. Sulhoff, “100 Gb/s error free transmission over 9,100 km using twenty 5 Gb/s WDM data channels,” in Optical Fiber Communication Conference Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD23.

N. S. Bergano and C. R. Davidson, “40 Gb/s WDM transmission of eight 5 Gb/s data channels over transoceanic distances using the conventional NRZ modulation format,” in Optical Fiber Communication Conference, Vol. 8 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), paper PD19.

Bronski, J. C.

J. C. Bronski and J. N. Kutz, “Modulational stability of plane waves in nonreturn-to-zero communications systems with dispersion management,” Opt. Lett. 21, 937–939 (1996).
[CrossRef] [PubMed]

J. C. Bronski and J. N. Kutz, “Asymptotic behavior of the nonlinear Schrödinger equation with rapidly varying, mean-zero dispersion,” Physica D (to be published).

J. C. Bronski and D. W. McLaughlin, “Semiclassical behavior in the NLS equation: optical shocks—focusing instabilities,” in Singular Limits of Dispersive Waves, N. M. Ercolani, ed. (Plenum, New York, 1994), pp. 21–38.

Chikama, T.

H. Onaka, H. Miyata, G. Ishikawa, K. Otsuka, H. Ooi, Y. Kai, S. Kinoshita, M. Seino, H. Nishimito, and T. Chikama, “1.1 Tb/s WDM transmission over a 150 km 1.3 µm zero-dispersion single-mode fiber,” in Optical Fiber Communication Conference, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD19.

Chraplyvy, A. R.

R. W. Tkach, A. R. Chraplyvy, F. Forguieri, A. H. Gnauck, and R. M. Derosier “4-Photon mixing and high-speed WDM systems,” J. Lightwave Technol. 13, 841–849 (1995).
[CrossRef]

A. R. Chraplyvy, A. H. Gnauck, R. W. Tkach, R. M. Derosier, C. R. Giles, B. M. Nyman, G. A. Ferguson, J. W. Sulhoff, and J. L. Zyskind, “1/3rd-Terabit/s transmission through 150-km of dispersion-managed fiber,” IEEE Photonics Technol. Lett. 7, 98–100 (1995).
[CrossRef]

A. H. Gnauck, A. R. Chraplyvy, R. W. Tkach, and R. M. Derosier, “160 Gbit/s (8×20 Gbit/s WDM) 300 km transmission with 50 km amplifier spacing and span-by-span dispersion reversal,” Electron. Lett. 30, 1241–1243 (1994).
[CrossRef]

A. H. Gnauck, A. R. Chraplyvy, R. W. Tkach, J. L. Zyskind, J. W. Sulhoff, A. J. Lucero, Y. Sun, R. M. Jopson, F. Forguieri, R. M. Derosier, C. Wolf, and A. R. McCormick, “One terabit/s transmission experiment,” in Optical Fiber Communication Conference, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD20.

Corbett, P. C.

N. S. Bergano, C. R. Davidson, D. L. Wilson, F. W. Kerfoot, M. D. Tremblay, M. D. Levonas, J. P. Morreale, J. D. Evankow, P. C. Corbett, M. A. Mills, G. A. Ferguson, A. M. Vengsarkar, J. R. Predrazzani, J. A. Nagel, J. L. Zyskind, and J. W. Sulhoff, “100 Gb/s error free transmission over 9,100 km using twenty 5 Gb/s WDM data channels,” in Optical Fiber Communication Conference Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD23.

Davidson, C. R.

N. S. Bergano, C. R. Davidson, D. L. Wilson, F. W. Kerfoot, M. D. Tremblay, M. D. Levonas, J. P. Morreale, J. D. Evankow, P. C. Corbett, M. A. Mills, G. A. Ferguson, A. M. Vengsarkar, J. R. Predrazzani, J. A. Nagel, J. L. Zyskind, and J. W. Sulhoff, “100 Gb/s error free transmission over 9,100 km using twenty 5 Gb/s WDM data channels,” in Optical Fiber Communication Conference Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD23.

N. S. Bergano and C. R. Davidson, “40 Gb/s WDM transmission of eight 5 Gb/s data channels over transoceanic distances using the conventional NRZ modulation format,” in Optical Fiber Communication Conference, Vol. 8 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), paper PD19.

Derosier, R. M.

A. R. Chraplyvy, A. H. Gnauck, R. W. Tkach, R. M. Derosier, C. R. Giles, B. M. Nyman, G. A. Ferguson, J. W. Sulhoff, and J. L. Zyskind, “1/3rd-Terabit/s transmission through 150-km of dispersion-managed fiber,” IEEE Photonics Technol. Lett. 7, 98–100 (1995).
[CrossRef]

R. W. Tkach, A. R. Chraplyvy, F. Forguieri, A. H. Gnauck, and R. M. Derosier “4-Photon mixing and high-speed WDM systems,” J. Lightwave Technol. 13, 841–849 (1995).
[CrossRef]

A. H. Gnauck, A. R. Chraplyvy, R. W. Tkach, and R. M. Derosier, “160 Gbit/s (8×20 Gbit/s WDM) 300 km transmission with 50 km amplifier spacing and span-by-span dispersion reversal,” Electron. Lett. 30, 1241–1243 (1994).
[CrossRef]

A. H. Gnauck, A. R. Chraplyvy, R. W. Tkach, J. L. Zyskind, J. W. Sulhoff, A. J. Lucero, Y. Sun, R. M. Jopson, F. Forguieri, R. M. Derosier, C. Wolf, and A. R. McCormick, “One terabit/s transmission experiment,” in Optical Fiber Communication Conference, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD20.

Evangelides, S. G.

L. F. Mollenauer, S. G. Evangelides, and J. P. Gordon, “Wavelength division multiplexing with solitons in ultra-long distance transmission using lumped amplifiers,” J. Lightwave Technol. 9, 362–367 (1991).
[CrossRef]

Evankow, J. D.

N. S. Bergano, C. R. Davidson, D. L. Wilson, F. W. Kerfoot, M. D. Tremblay, M. D. Levonas, J. P. Morreale, J. D. Evankow, P. C. Corbett, M. A. Mills, G. A. Ferguson, A. M. Vengsarkar, J. R. Predrazzani, J. A. Nagel, J. L. Zyskind, and J. W. Sulhoff, “100 Gb/s error free transmission over 9,100 km using twenty 5 Gb/s WDM data channels,” in Optical Fiber Communication Conference Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD23.

Ferguson, G. A.

A. R. Chraplyvy, A. H. Gnauck, R. W. Tkach, R. M. Derosier, C. R. Giles, B. M. Nyman, G. A. Ferguson, J. W. Sulhoff, and J. L. Zyskind, “1/3rd-Terabit/s transmission through 150-km of dispersion-managed fiber,” IEEE Photonics Technol. Lett. 7, 98–100 (1995).
[CrossRef]

N. S. Bergano, C. R. Davidson, D. L. Wilson, F. W. Kerfoot, M. D. Tremblay, M. D. Levonas, J. P. Morreale, J. D. Evankow, P. C. Corbett, M. A. Mills, G. A. Ferguson, A. M. Vengsarkar, J. R. Predrazzani, J. A. Nagel, J. L. Zyskind, and J. W. Sulhoff, “100 Gb/s error free transmission over 9,100 km using twenty 5 Gb/s WDM data channels,” in Optical Fiber Communication Conference Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD23.

Forguieri, F.

R. W. Tkach, A. R. Chraplyvy, F. Forguieri, A. H. Gnauck, and R. M. Derosier “4-Photon mixing and high-speed WDM systems,” J. Lightwave Technol. 13, 841–849 (1995).
[CrossRef]

A. H. Gnauck, A. R. Chraplyvy, R. W. Tkach, J. L. Zyskind, J. W. Sulhoff, A. J. Lucero, Y. Sun, R. M. Jopson, F. Forguieri, R. M. Derosier, C. Wolf, and A. R. McCormick, “One terabit/s transmission experiment,” in Optical Fiber Communication Conference, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD20.

Gabitov, I. R.

Giles, C. R.

A. R. Chraplyvy, A. H. Gnauck, R. W. Tkach, R. M. Derosier, C. R. Giles, B. M. Nyman, G. A. Ferguson, J. W. Sulhoff, and J. L. Zyskind, “1/3rd-Terabit/s transmission through 150-km of dispersion-managed fiber,” IEEE Photonics Technol. Lett. 7, 98–100 (1995).
[CrossRef]

Gnauck, A. H.

A. R. Chraplyvy, A. H. Gnauck, R. W. Tkach, R. M. Derosier, C. R. Giles, B. M. Nyman, G. A. Ferguson, J. W. Sulhoff, and J. L. Zyskind, “1/3rd-Terabit/s transmission through 150-km of dispersion-managed fiber,” IEEE Photonics Technol. Lett. 7, 98–100 (1995).
[CrossRef]

R. W. Tkach, A. R. Chraplyvy, F. Forguieri, A. H. Gnauck, and R. M. Derosier “4-Photon mixing and high-speed WDM systems,” J. Lightwave Technol. 13, 841–849 (1995).
[CrossRef]

A. H. Gnauck, A. R. Chraplyvy, R. W. Tkach, and R. M. Derosier, “160 Gbit/s (8×20 Gbit/s WDM) 300 km transmission with 50 km amplifier spacing and span-by-span dispersion reversal,” Electron. Lett. 30, 1241–1243 (1994).
[CrossRef]

A. H. Gnauck, A. R. Chraplyvy, R. W. Tkach, J. L. Zyskind, J. W. Sulhoff, A. J. Lucero, Y. Sun, R. M. Jopson, F. Forguieri, R. M. Derosier, C. Wolf, and A. R. McCormick, “One terabit/s transmission experiment,” in Optical Fiber Communication Conference, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD20.

Gordon, J. P.

L. F. Mollenauer, S. G. Evangelides, and J. P. Gordon, “Wavelength division multiplexing with solitons in ultra-long distance transmission using lumped amplifiers,” J. Lightwave Technol. 9, 362–367 (1991).
[CrossRef]

Grishkowsky, D.

J. E. Rothenberg and D. Grishkowsky, “Observation of the formation of an optical-intensity shock and wave breaking in the nonlinear propagation of pulses in optical fibers,” Phys. Rev. Lett. 62, 531–534 (1989).
[CrossRef] [PubMed]

Harvey, G. T.

L. F. Mollenauer, E. Lichtman, G. T. Harvey, M. J. Neubelt, and B. M. Nyman, “Demonstration of error-free soliton transmission over more than 15000 km at 2.5 Gbit/s, single-channel, and over more than 11000 km at 10 Gbit/s in 2-channel WDM,” Electron. Lett. 28, 792–794 (1992).
[CrossRef]

Hasegawa, A.

Hawkins, R. J.

A. M. Weiner, W. J. Tomlinson, R. N. Thurston, D. E. Leaird, J. P. Heritage, E. M. Kirschner, and R. J. Hawkins, “Experimental-observation of the fundamental dark soliton in optical fibers,” Phys. Rev. Lett. 61, 2445–2448 (1988).
[CrossRef] [PubMed]

Heritage, J. P.

A. M. Weiner, W. J. Tomlinson, R. N. Thurston, D. E. Leaird, J. P. Heritage, E. M. Kirschner, and R. J. Hawkins, “Experimental-observation of the fundamental dark soliton in optical fibers,” Phys. Rev. Lett. 61, 2445–2448 (1988).
[CrossRef] [PubMed]

Hou, T. Y.

T. Y. Hou, J. S. Lowengrub, and M. J. Shelley, “Removing the stiffness from interfacial flows with surface tension,” J. Comput. Phys. 114, 312–338 (1994).
[CrossRef]

Ishikawa, G.

H. Onaka, H. Miyata, G. Ishikawa, K. Otsuka, H. Ooi, Y. Kai, S. Kinoshita, M. Seino, H. Nishimito, and T. Chikama, “1.1 Tb/s WDM transmission over a 150 km 1.3 µm zero-dispersion single-mode fiber,” in Optical Fiber Communication Conference, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD19.

Jain, M.

N. Tzoar and M. Jain, “Self-phase modulation in long-geometry optical waveguides,” Phys. Rev. A 23, 1266–1270 (1981).
[CrossRef]

Jopson, R. M.

A. H. Gnauck, A. R. Chraplyvy, R. W. Tkach, J. L. Zyskind, J. W. Sulhoff, A. J. Lucero, Y. Sun, R. M. Jopson, F. Forguieri, R. M. Derosier, C. Wolf, and A. R. McCormick, “One terabit/s transmission experiment,” in Optical Fiber Communication Conference, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD20.

Kai, Y.

H. Onaka, H. Miyata, G. Ishikawa, K. Otsuka, H. Ooi, Y. Kai, S. Kinoshita, M. Seino, H. Nishimito, and T. Chikama, “1.1 Tb/s WDM transmission over a 150 km 1.3 µm zero-dispersion single-mode fiber,” in Optical Fiber Communication Conference, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD19.

Kerfoot, F. W.

N. S. Bergano, C. R. Davidson, D. L. Wilson, F. W. Kerfoot, M. D. Tremblay, M. D. Levonas, J. P. Morreale, J. D. Evankow, P. C. Corbett, M. A. Mills, G. A. Ferguson, A. M. Vengsarkar, J. R. Predrazzani, J. A. Nagel, J. L. Zyskind, and J. W. Sulhoff, “100 Gb/s error free transmission over 9,100 km using twenty 5 Gb/s WDM data channels,” in Optical Fiber Communication Conference Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD23.

Kimura, Y.

M. Nakazawa, K. Susuki, E. Yamada, H. Kubota, Y. Kimura, and M. Takaya, “Experimental demonstration of soliton data-transmission over unlimited distances with soliton control in time and frequency domains,” Electron. Lett. 29, 729–730 (1993).
[CrossRef]

Kinoshita, S.

H. Onaka, H. Miyata, G. Ishikawa, K. Otsuka, H. Ooi, Y. Kai, S. Kinoshita, M. Seino, H. Nishimito, and T. Chikama, “1.1 Tb/s WDM transmission over a 150 km 1.3 µm zero-dispersion single-mode fiber,” in Optical Fiber Communication Conference, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD19.

Kirschner, E. M.

A. M. Weiner, W. J. Tomlinson, R. N. Thurston, D. E. Leaird, J. P. Heritage, E. M. Kirschner, and R. J. Hawkins, “Experimental-observation of the fundamental dark soliton in optical fibers,” Phys. Rev. Lett. 61, 2445–2448 (1988).
[CrossRef] [PubMed]

Kivshar, Y. S.

K. H. Spatshek, S. K. Turitsyn, and Y. S. Kivshar, “Average envelope soliton dynamics in systems with periodically varying dispersion,” Phys. Lett. A 204, 269–273 (1995).
[CrossRef]

Kodama, Y.

Kubota, H.

M. Nakazawa, K. Susuki, E. Yamada, H. Kubota, Y. Kimura, and M. Takaya, “Experimental demonstration of soliton data-transmission over unlimited distances with soliton control in time and frequency domains,” Electron. Lett. 29, 729–730 (1993).
[CrossRef]

Kurtzke, C.

C. Kurtzke, “Suppression of fiber nonlinearities by appropriate dispersion management,” IEEE Photonics Technol. Lett. 5, 1250–1253 (1993).
[CrossRef]

Kutz, J. N.

J. C. Bronski and J. N. Kutz, “Modulational stability of plane waves in nonreturn-to-zero communications systems with dispersion management,” Opt. Lett. 21, 937–939 (1996).
[CrossRef] [PubMed]

J. C. Bronski and J. N. Kutz, “Asymptotic behavior of the nonlinear Schrödinger equation with rapidly varying, mean-zero dispersion,” Physica D (to be published).

Leaird, D. E.

A. M. Weiner, W. J. Tomlinson, R. N. Thurston, D. E. Leaird, J. P. Heritage, E. M. Kirschner, and R. J. Hawkins, “Experimental-observation of the fundamental dark soliton in optical fibers,” Phys. Rev. Lett. 61, 2445–2448 (1988).
[CrossRef] [PubMed]

Levonas, M. D.

N. S. Bergano, C. R. Davidson, D. L. Wilson, F. W. Kerfoot, M. D. Tremblay, M. D. Levonas, J. P. Morreale, J. D. Evankow, P. C. Corbett, M. A. Mills, G. A. Ferguson, A. M. Vengsarkar, J. R. Predrazzani, J. A. Nagel, J. L. Zyskind, and J. W. Sulhoff, “100 Gb/s error free transmission over 9,100 km using twenty 5 Gb/s WDM data channels,” in Optical Fiber Communication Conference Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD23.

Lichtman, E.

L. F. Mollenauer, E. Lichtman, G. T. Harvey, M. J. Neubelt, and B. M. Nyman, “Demonstration of error-free soliton transmission over more than 15000 km at 2.5 Gbit/s, single-channel, and over more than 11000 km at 10 Gbit/s in 2-channel WDM,” Electron. Lett. 28, 792–794 (1992).
[CrossRef]

Lowengrub, J. S.

T. Y. Hou, J. S. Lowengrub, and M. J. Shelley, “Removing the stiffness from interfacial flows with surface tension,” J. Comput. Phys. 114, 312–338 (1994).
[CrossRef]

Lucero, A. J.

A. H. Gnauck, A. R. Chraplyvy, R. W. Tkach, J. L. Zyskind, J. W. Sulhoff, A. J. Lucero, Y. Sun, R. M. Jopson, F. Forguieri, R. M. Derosier, C. Wolf, and A. R. McCormick, “One terabit/s transmission experiment,” in Optical Fiber Communication Conference, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD20.

McCormick, A. R.

A. H. Gnauck, A. R. Chraplyvy, R. W. Tkach, J. L. Zyskind, J. W. Sulhoff, A. J. Lucero, Y. Sun, R. M. Jopson, F. Forguieri, R. M. Derosier, C. Wolf, and A. R. McCormick, “One terabit/s transmission experiment,” in Optical Fiber Communication Conference, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD20.

McLaughlin, D. W.

J. C. Bronski and D. W. McLaughlin, “Semiclassical behavior in the NLS equation: optical shocks—focusing instabilities,” in Singular Limits of Dispersive Waves, N. M. Ercolani, ed. (Plenum, New York, 1994), pp. 21–38.

Mills, M. A.

N. S. Bergano, C. R. Davidson, D. L. Wilson, F. W. Kerfoot, M. D. Tremblay, M. D. Levonas, J. P. Morreale, J. D. Evankow, P. C. Corbett, M. A. Mills, G. A. Ferguson, A. M. Vengsarkar, J. R. Predrazzani, J. A. Nagel, J. L. Zyskind, and J. W. Sulhoff, “100 Gb/s error free transmission over 9,100 km using twenty 5 Gb/s WDM data channels,” in Optical Fiber Communication Conference Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD23.

Miyata, H.

H. Onaka, H. Miyata, G. Ishikawa, K. Otsuka, H. Ooi, Y. Kai, S. Kinoshita, M. Seino, H. Nishimito, and T. Chikama, “1.1 Tb/s WDM transmission over a 150 km 1.3 µm zero-dispersion single-mode fiber,” in Optical Fiber Communication Conference, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD19.

Mollenauer, L. F.

L. F. Mollenauer, E. Lichtman, G. T. Harvey, M. J. Neubelt, and B. M. Nyman, “Demonstration of error-free soliton transmission over more than 15000 km at 2.5 Gbit/s, single-channel, and over more than 11000 km at 10 Gbit/s in 2-channel WDM,” Electron. Lett. 28, 792–794 (1992).
[CrossRef]

L. F. Mollenauer, S. G. Evangelides, and J. P. Gordon, “Wavelength division multiplexing with solitons in ultra-long distance transmission using lumped amplifiers,” J. Lightwave Technol. 9, 362–367 (1991).
[CrossRef]

Morreale, J. P.

N. S. Bergano, C. R. Davidson, D. L. Wilson, F. W. Kerfoot, M. D. Tremblay, M. D. Levonas, J. P. Morreale, J. D. Evankow, P. C. Corbett, M. A. Mills, G. A. Ferguson, A. M. Vengsarkar, J. R. Predrazzani, J. A. Nagel, J. L. Zyskind, and J. W. Sulhoff, “100 Gb/s error free transmission over 9,100 km using twenty 5 Gb/s WDM data channels,” in Optical Fiber Communication Conference Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD23.

Nagel, J. A.

N. S. Bergano, C. R. Davidson, D. L. Wilson, F. W. Kerfoot, M. D. Tremblay, M. D. Levonas, J. P. Morreale, J. D. Evankow, P. C. Corbett, M. A. Mills, G. A. Ferguson, A. M. Vengsarkar, J. R. Predrazzani, J. A. Nagel, J. L. Zyskind, and J. W. Sulhoff, “100 Gb/s error free transmission over 9,100 km using twenty 5 Gb/s WDM data channels,” in Optical Fiber Communication Conference Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD23.

Nakazawa, M.

M. Nakazawa, K. Susuki, E. Yamada, H. Kubota, Y. Kimura, and M. Takaya, “Experimental demonstration of soliton data-transmission over unlimited distances with soliton control in time and frequency domains,” Electron. Lett. 29, 729–730 (1993).
[CrossRef]

Neubelt, M. J.

L. F. Mollenauer, E. Lichtman, G. T. Harvey, M. J. Neubelt, and B. M. Nyman, “Demonstration of error-free soliton transmission over more than 15000 km at 2.5 Gbit/s, single-channel, and over more than 11000 km at 10 Gbit/s in 2-channel WDM,” Electron. Lett. 28, 792–794 (1992).
[CrossRef]

Nishimito, H.

H. Onaka, H. Miyata, G. Ishikawa, K. Otsuka, H. Ooi, Y. Kai, S. Kinoshita, M. Seino, H. Nishimito, and T. Chikama, “1.1 Tb/s WDM transmission over a 150 km 1.3 µm zero-dispersion single-mode fiber,” in Optical Fiber Communication Conference, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD19.

Nyman, B. M.

A. R. Chraplyvy, A. H. Gnauck, R. W. Tkach, R. M. Derosier, C. R. Giles, B. M. Nyman, G. A. Ferguson, J. W. Sulhoff, and J. L. Zyskind, “1/3rd-Terabit/s transmission through 150-km of dispersion-managed fiber,” IEEE Photonics Technol. Lett. 7, 98–100 (1995).
[CrossRef]

L. F. Mollenauer, E. Lichtman, G. T. Harvey, M. J. Neubelt, and B. M. Nyman, “Demonstration of error-free soliton transmission over more than 15000 km at 2.5 Gbit/s, single-channel, and over more than 11000 km at 10 Gbit/s in 2-channel WDM,” Electron. Lett. 28, 792–794 (1992).
[CrossRef]

Onaka, H.

H. Onaka, H. Miyata, G. Ishikawa, K. Otsuka, H. Ooi, Y. Kai, S. Kinoshita, M. Seino, H. Nishimito, and T. Chikama, “1.1 Tb/s WDM transmission over a 150 km 1.3 µm zero-dispersion single-mode fiber,” in Optical Fiber Communication Conference, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD19.

Ooi, H.

H. Onaka, H. Miyata, G. Ishikawa, K. Otsuka, H. Ooi, Y. Kai, S. Kinoshita, M. Seino, H. Nishimito, and T. Chikama, “1.1 Tb/s WDM transmission over a 150 km 1.3 µm zero-dispersion single-mode fiber,” in Optical Fiber Communication Conference, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD19.

Orszag, S. A.

C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978), Chap. 7.

Otsuka, K.

H. Onaka, H. Miyata, G. Ishikawa, K. Otsuka, H. Ooi, Y. Kai, S. Kinoshita, M. Seino, H. Nishimito, and T. Chikama, “1.1 Tb/s WDM transmission over a 150 km 1.3 µm zero-dispersion single-mode fiber,” in Optical Fiber Communication Conference, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD19.

Predrazzani, J. R.

N. S. Bergano, C. R. Davidson, D. L. Wilson, F. W. Kerfoot, M. D. Tremblay, M. D. Levonas, J. P. Morreale, J. D. Evankow, P. C. Corbett, M. A. Mills, G. A. Ferguson, A. M. Vengsarkar, J. R. Predrazzani, J. A. Nagel, J. L. Zyskind, and J. W. Sulhoff, “100 Gb/s error free transmission over 9,100 km using twenty 5 Gb/s WDM data channels,” in Optical Fiber Communication Conference Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD23.

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J. E. Rothenberg, “Femtosecond optical shocks and wave breaking in fiber propagation,” J. Opt. Soc. Am. B 6, 2392–2401 (1989).
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J. E. Rothenberg and D. Grishkowsky, “Observation of the formation of an optical-intensity shock and wave breaking in the nonlinear propagation of pulses in optical fibers,” Phys. Rev. Lett. 62, 531–534 (1989).
[CrossRef] [PubMed]

Seino, M.

H. Onaka, H. Miyata, G. Ishikawa, K. Otsuka, H. Ooi, Y. Kai, S. Kinoshita, M. Seino, H. Nishimito, and T. Chikama, “1.1 Tb/s WDM transmission over a 150 km 1.3 µm zero-dispersion single-mode fiber,” in Optical Fiber Communication Conference, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD19.

Shelley, M. J.

T. Y. Hou, J. S. Lowengrub, and M. J. Shelley, “Removing the stiffness from interfacial flows with surface tension,” J. Comput. Phys. 114, 312–338 (1994).
[CrossRef]

Spatshek, K. H.

K. H. Spatshek, S. K. Turitsyn, and Y. S. Kivshar, “Average envelope soliton dynamics in systems with periodically varying dispersion,” Phys. Lett. A 204, 269–273 (1995).
[CrossRef]

Sulhoff, J. W.

A. R. Chraplyvy, A. H. Gnauck, R. W. Tkach, R. M. Derosier, C. R. Giles, B. M. Nyman, G. A. Ferguson, J. W. Sulhoff, and J. L. Zyskind, “1/3rd-Terabit/s transmission through 150-km of dispersion-managed fiber,” IEEE Photonics Technol. Lett. 7, 98–100 (1995).
[CrossRef]

A. H. Gnauck, A. R. Chraplyvy, R. W. Tkach, J. L. Zyskind, J. W. Sulhoff, A. J. Lucero, Y. Sun, R. M. Jopson, F. Forguieri, R. M. Derosier, C. Wolf, and A. R. McCormick, “One terabit/s transmission experiment,” in Optical Fiber Communication Conference, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD20.

N. S. Bergano, C. R. Davidson, D. L. Wilson, F. W. Kerfoot, M. D. Tremblay, M. D. Levonas, J. P. Morreale, J. D. Evankow, P. C. Corbett, M. A. Mills, G. A. Ferguson, A. M. Vengsarkar, J. R. Predrazzani, J. A. Nagel, J. L. Zyskind, and J. W. Sulhoff, “100 Gb/s error free transmission over 9,100 km using twenty 5 Gb/s WDM data channels,” in Optical Fiber Communication Conference Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD23.

Sun, Y.

A. H. Gnauck, A. R. Chraplyvy, R. W. Tkach, J. L. Zyskind, J. W. Sulhoff, A. J. Lucero, Y. Sun, R. M. Jopson, F. Forguieri, R. M. Derosier, C. Wolf, and A. R. McCormick, “One terabit/s transmission experiment,” in Optical Fiber Communication Conference, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD20.

Susuki, K.

M. Nakazawa, K. Susuki, E. Yamada, H. Kubota, Y. Kimura, and M. Takaya, “Experimental demonstration of soliton data-transmission over unlimited distances with soliton control in time and frequency domains,” Electron. Lett. 29, 729–730 (1993).
[CrossRef]

Taha, T. R.

T. R. Taha and M. J. Ablowitz, “Analytical and numerical aspects of certain nonlinear evolution equations: II. Numerical, nonlinear Schrödinger equation,” J. Comput. Phys. 55, 203–230 (1984).
[CrossRef]

Takaya, M.

M. Nakazawa, K. Susuki, E. Yamada, H. Kubota, Y. Kimura, and M. Takaya, “Experimental demonstration of soliton data-transmission over unlimited distances with soliton control in time and frequency domains,” Electron. Lett. 29, 729–730 (1993).
[CrossRef]

Thurston, R. N.

A. M. Weiner, W. J. Tomlinson, R. N. Thurston, D. E. Leaird, J. P. Heritage, E. M. Kirschner, and R. J. Hawkins, “Experimental-observation of the fundamental dark soliton in optical fibers,” Phys. Rev. Lett. 61, 2445–2448 (1988).
[CrossRef] [PubMed]

Tkach, R. W.

R. W. Tkach, A. R. Chraplyvy, F. Forguieri, A. H. Gnauck, and R. M. Derosier “4-Photon mixing and high-speed WDM systems,” J. Lightwave Technol. 13, 841–849 (1995).
[CrossRef]

A. R. Chraplyvy, A. H. Gnauck, R. W. Tkach, R. M. Derosier, C. R. Giles, B. M. Nyman, G. A. Ferguson, J. W. Sulhoff, and J. L. Zyskind, “1/3rd-Terabit/s transmission through 150-km of dispersion-managed fiber,” IEEE Photonics Technol. Lett. 7, 98–100 (1995).
[CrossRef]

A. H. Gnauck, A. R. Chraplyvy, R. W. Tkach, and R. M. Derosier, “160 Gbit/s (8×20 Gbit/s WDM) 300 km transmission with 50 km amplifier spacing and span-by-span dispersion reversal,” Electron. Lett. 30, 1241–1243 (1994).
[CrossRef]

A. H. Gnauck, A. R. Chraplyvy, R. W. Tkach, J. L. Zyskind, J. W. Sulhoff, A. J. Lucero, Y. Sun, R. M. Jopson, F. Forguieri, R. M. Derosier, C. Wolf, and A. R. McCormick, “One terabit/s transmission experiment,” in Optical Fiber Communication Conference, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD20.

Tomlinson, W. J.

A. M. Weiner, W. J. Tomlinson, R. N. Thurston, D. E. Leaird, J. P. Heritage, E. M. Kirschner, and R. J. Hawkins, “Experimental-observation of the fundamental dark soliton in optical fibers,” Phys. Rev. Lett. 61, 2445–2448 (1988).
[CrossRef] [PubMed]

Tremblay, M. D.

N. S. Bergano, C. R. Davidson, D. L. Wilson, F. W. Kerfoot, M. D. Tremblay, M. D. Levonas, J. P. Morreale, J. D. Evankow, P. C. Corbett, M. A. Mills, G. A. Ferguson, A. M. Vengsarkar, J. R. Predrazzani, J. A. Nagel, J. L. Zyskind, and J. W. Sulhoff, “100 Gb/s error free transmission over 9,100 km using twenty 5 Gb/s WDM data channels,” in Optical Fiber Communication Conference Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD23.

Turitsyn, S. K.

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

K. H. Spatshek, S. K. Turitsyn, and Y. S. Kivshar, “Average envelope soliton dynamics in systems with periodically varying dispersion,” Phys. Lett. A 204, 269–273 (1995).
[CrossRef]

Tzoar, N.

N. Tzoar and M. Jain, “Self-phase modulation in long-geometry optical waveguides,” Phys. Rev. A 23, 1266–1270 (1981).
[CrossRef]

Vengsarkar, A. M.

N. S. Bergano, C. R. Davidson, D. L. Wilson, F. W. Kerfoot, M. D. Tremblay, M. D. Levonas, J. P. Morreale, J. D. Evankow, P. C. Corbett, M. A. Mills, G. A. Ferguson, A. M. Vengsarkar, J. R. Predrazzani, J. A. Nagel, J. L. Zyskind, and J. W. Sulhoff, “100 Gb/s error free transmission over 9,100 km using twenty 5 Gb/s WDM data channels,” in Optical Fiber Communication Conference Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD23.

Wabnitz, S.

Weiner, A. M.

A. M. Weiner, W. J. Tomlinson, R. N. Thurston, D. E. Leaird, J. P. Heritage, E. M. Kirschner, and R. J. Hawkins, “Experimental-observation of the fundamental dark soliton in optical fibers,” Phys. Rev. Lett. 61, 2445–2448 (1988).
[CrossRef] [PubMed]

Wilson, D. L.

N. S. Bergano, C. R. Davidson, D. L. Wilson, F. W. Kerfoot, M. D. Tremblay, M. D. Levonas, J. P. Morreale, J. D. Evankow, P. C. Corbett, M. A. Mills, G. A. Ferguson, A. M. Vengsarkar, J. R. Predrazzani, J. A. Nagel, J. L. Zyskind, and J. W. Sulhoff, “100 Gb/s error free transmission over 9,100 km using twenty 5 Gb/s WDM data channels,” in Optical Fiber Communication Conference Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD23.

Wolf, C.

A. H. Gnauck, A. R. Chraplyvy, R. W. Tkach, J. L. Zyskind, J. W. Sulhoff, A. J. Lucero, Y. Sun, R. M. Jopson, F. Forguieri, R. M. Derosier, C. Wolf, and A. R. McCormick, “One terabit/s transmission experiment,” in Optical Fiber Communication Conference, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD20.

Yamada, E.

M. Nakazawa, K. Susuki, E. Yamada, H. Kubota, Y. Kimura, and M. Takaya, “Experimental demonstration of soliton data-transmission over unlimited distances with soliton control in time and frequency domains,” Electron. Lett. 29, 729–730 (1993).
[CrossRef]

Zauderer, E.

E. Zauderer, Partial Differential Equations of Applied Mathematics (Wiley, 1989), Chap. 3, pp. 136–139.

Zyskind, J. L.

A. R. Chraplyvy, A. H. Gnauck, R. W. Tkach, R. M. Derosier, C. R. Giles, B. M. Nyman, G. A. Ferguson, J. W. Sulhoff, and J. L. Zyskind, “1/3rd-Terabit/s transmission through 150-km of dispersion-managed fiber,” IEEE Photonics Technol. Lett. 7, 98–100 (1995).
[CrossRef]

A. H. Gnauck, A. R. Chraplyvy, R. W. Tkach, J. L. Zyskind, J. W. Sulhoff, A. J. Lucero, Y. Sun, R. M. Jopson, F. Forguieri, R. M. Derosier, C. Wolf, and A. R. McCormick, “One terabit/s transmission experiment,” in Optical Fiber Communication Conference, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD20.

N. S. Bergano, C. R. Davidson, D. L. Wilson, F. W. Kerfoot, M. D. Tremblay, M. D. Levonas, J. P. Morreale, J. D. Evankow, P. C. Corbett, M. A. Mills, G. A. Ferguson, A. M. Vengsarkar, J. R. Predrazzani, J. A. Nagel, J. L. Zyskind, and J. W. Sulhoff, “100 Gb/s error free transmission over 9,100 km using twenty 5 Gb/s WDM data channels,” in Optical Fiber Communication Conference Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD23.

Electron. Lett. (3)

A. H. Gnauck, A. R. Chraplyvy, R. W. Tkach, and R. M. Derosier, “160 Gbit/s (8×20 Gbit/s WDM) 300 km transmission with 50 km amplifier spacing and span-by-span dispersion reversal,” Electron. Lett. 30, 1241–1243 (1994).
[CrossRef]

L. F. Mollenauer, E. Lichtman, G. T. Harvey, M. J. Neubelt, and B. M. Nyman, “Demonstration of error-free soliton transmission over more than 15000 km at 2.5 Gbit/s, single-channel, and over more than 11000 km at 10 Gbit/s in 2-channel WDM,” Electron. Lett. 28, 792–794 (1992).
[CrossRef]

M. Nakazawa, K. Susuki, E. Yamada, H. Kubota, Y. Kimura, and M. Takaya, “Experimental demonstration of soliton data-transmission over unlimited distances with soliton control in time and frequency domains,” Electron. Lett. 29, 729–730 (1993).
[CrossRef]

IEEE Photonics Technol. Lett. (2)

A. R. Chraplyvy, A. H. Gnauck, R. W. Tkach, R. M. Derosier, C. R. Giles, B. M. Nyman, G. A. Ferguson, J. W. Sulhoff, and J. L. Zyskind, “1/3rd-Terabit/s transmission through 150-km of dispersion-managed fiber,” IEEE Photonics Technol. Lett. 7, 98–100 (1995).
[CrossRef]

C. Kurtzke, “Suppression of fiber nonlinearities by appropriate dispersion management,” IEEE Photonics Technol. Lett. 5, 1250–1253 (1993).
[CrossRef]

J. Comput. Phys. (2)

T. Y. Hou, J. S. Lowengrub, and M. J. Shelley, “Removing the stiffness from interfacial flows with surface tension,” J. Comput. Phys. 114, 312–338 (1994).
[CrossRef]

T. R. Taha and M. J. Ablowitz, “Analytical and numerical aspects of certain nonlinear evolution equations: II. Numerical, nonlinear Schrödinger equation,” J. Comput. Phys. 55, 203–230 (1984).
[CrossRef]

J. Lightwave Technol. (2)

L. F. Mollenauer, S. G. Evangelides, and J. P. Gordon, “Wavelength division multiplexing with solitons in ultra-long distance transmission using lumped amplifiers,” J. Lightwave Technol. 9, 362–367 (1991).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Typical NRZ bit-stream pattern.

Fig. 2
Fig. 2

Approximate dispersion compensation schemes employed in (a) Ref. 3 and (b) Ref. 2. Each scheme is such that σ(Z)=0 over a given period P.

Fig. 3
Fig. 3

Typical pulse evolution over 9600 km as governed by Eq. (1) for the dispersion map shown in Fig. 2(a). In (a) we consider the pulse just after the anomalous-dispersion fiber segment. In (b) we consider the pulse just after the normal-dispersion segment. Note (a) the pulse compression and (b) the pulse spreading.

Fig. 4
Fig. 4

Comparison of the effective evolution equations and full governing equation (1) for the case in which k1>0. Note the good agreement of the pulse spreading for distances up to 4800 km.

Fig. 5
Fig. 5

Comparison of the effective evolution equations and full governing equation (1) for the case in which k1<0. Note the good agreement of the pulse compression for distances up to 4800 km.

Fig. 6
Fig. 6

Comparison of the effective evolution equations and full governing equation (1) for the case in which k1=0 (symmetrized dispersion map). Note the minimal deformations and the good agreement of the pulse propagation for distances up to 9600 km.

Fig. 7
Fig. 7

Normalized error measurements between the full numerical solution of Eq. (1) and the leading order and the effective evolutions. Note that the effective evolution error is an order of magnitude smaller than that corresponding to the leading-order behavior.

Fig. 8
Fig. 8

Long-term behavior of the full numerical solutions develops a wave-breaking (shock) structure at the fronts of the pulses. Note that this shock structure grows for longer propagation lengths. It is this mechanism that leads to the closing of the eye diagram.

Equations (53)

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iQZ+σ(Z)22QT2+|Q|2Q+iγ(Z)Q=0,
ZNL=λ0Aeff2πn2|E0|2=4016km,
γ(Z)Q=ΓQ-n=1N{[exp(Γln-1)]δ(Z-nln)}Q.
σ(Z)=β+β-0<Z<Z+Z+<Z<P=Z++Z-,
β±=λ02ZNL2πcT02D±,
σ(Z)=1P0Pσ(Z)dZ=β+Z++β-Z-P=0.
=PZNL1.
σ(Z)-14.2214.220<Z<0.50.5<Z<,
σ(Z)-11.3796.690<Z<0.8950.895<Z<,
Q(Z, T)=a(Z)U(Z, T),
iUZ+σ(Z)22UT2+[1+g(Z)]|U|2U=0.
g(Z)=2Γl exp(-2ΓZ)1-exp(-2Γl)-1.
U=V+O().
dVdZ=i|V|2V+k1V22V*T2+2VT2V+VT2V*,
k1=β¯l81+21-exp(-Γl)1-exp(-2Γl)O(β¯),
V(Z, T)=R(Z, T) exp[iӨ(Z, T)].
RZ=k1TRRT,
ӨZ=-k1R2ӨT2-R.
RdT=|Q|2dT=const,
ӨTRdT=iQQT*-Q*QTdT=const.
Zbreak=2πcT02λ02(|D+|Z++|D-|Z-)1/2.
RZ=-|k1|22T21+|k1|22T2R2.
ӨZ=-k1R2T21+k12T2Ө-R
Q(0, T)=(A/2){tanh[W(T+Tf)]+tanh[-W(T-Tf)]},
E=|U(Z, T)|-|Uasymptotic||U(Z, 0)|,
UZ=[1+g(Z)]X0+σ(Z)X0D,
X0=i|U|2U,
X0D=i22UT2.
U=exp(Φ·)V=[1+Φ·+1/2(Φ·)2+]V,
Φ·=ϕV+ϕTVT+2ϕT22VT2++c.c.,
U=exp(Φ·)V=V+ϕ+½(Φ·ϕ)+.
dVdZ=Y,
ddZ=Z+dVdZ·.
ϕ=0+ϕ1+ϕ2+,
Y=X0+Y1+Y2+,
dϕ1dZ=σ(Z)X0D+g(Z)X0,
ϕ1=[μ(Z)X0D+g˜(Z)X0]+ϕ10.
dμdZ=σ,dg˜dZ=g,
dϕ2dZ=[ϕ1, X0]+12ϕ1, dϕ1dZ-Y1,
Y1=[ϕ10, X0]+k1[X0D, X0],
k1=½μg-σg˜.
ϕ2=2β(Z)[X0D, X0]+ϕ20,
dβdZ=μ+gμ-g˜σ2,
dϕ3dZ=[ϕ2, X0]+12ϕ2,dϕ1dZ+112ϕ1, ϕ1,dϕ1dZ-Y2.
Y2=[ϕ20, X0]+2k2[[X0D, X0], X0]+2k3[[X0D, X0], X0D],
k2=β+βa2-g˜(gμ-σg˜)12,
k3=βσ2-μ(gμ-σg˜)12.
ϕ20=2k2[X0D, X0],
Y2=2k3[[X0D, X0], X0D],
dVdZ=i|V|2V+k1[X0D, X0]+2k3[[X0D, X0], X0D]+,
U=V+ϕ1+½Φ1·ϕ1+ϕ2+O(3),
[X0D, X0]=V22V*T2+2VT2V+VT2V*,
[[X0D, X0],X0D]=iV24V*T4+2VT22V*+2V2VT22+4VVT3V*T3+42VT2VT2+4VT22V*T2.

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