Abstract

The propagation of nonreturn-to-zero pulses, composed by a superposition of two exact shock-wave solutions of a complex-cubic Ginzburg–Landau equation linearly coupled to a linear nondispersive equation, is studied in detail. The model describes the distributed (average) propagation in a dual-core erbium-doped fiber-amplifier-supported optical-fiber system where stabilization is achieved by means of short segments of an extra lossy core that is parallel and coupled to the main one. The linear-stability analysis of the two asymptotic states of the shock wave in combination with direct numerical simulations provide necessary conditions for optimal propagation of the nonreturn-to-zero pulse. The enhancement of the propagation distance by at least an order of magnitude, under a suitable choice of the parameters, establishes the beneficial role of the passive channel.

© 2002 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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  27. J. R. Simpson, H. T. Shang, L. F. Mollenauer, N. A. Olsson, P. C. Becker, K. S. Kranz, P. J. Lemaire, and M. J. Neubelt, “Performance of a distributed erbium-doped dispersion-shifted fiber amplifier,” J. Lightwave Technol. 9, 228–233 (1991).
    [CrossRef]
  28. J. Atai and B. A. Malomed, “Exact stable pulses in asymmetric linearly coupled Ginzburg–Landau equations,” Phys. Lett. A 246, 412–422 (1998).
    [CrossRef]
  29. A. Hasegawa, “Generation of a train of soliton pulses by induced modulational instability in optical fibers,” Opt. Lett. 9, 288–290 (1984).
    [CrossRef] [PubMed]
  30. K. Tai, A. Tomita, and A. Hasegawa, “Observation of modulational instability in optical fibers,” Phys. Rev. Lett. 56, 135–138 (1986).
    [CrossRef] [PubMed]

2000 (2)

N. Efremidis, K. Hizanidis, B. A. Malomed, H. E. Nistazakis, and D. J. Frantzeskakis, “Stabilizing the Pereira–Stenflo solitons in nonlinear optical fibers,” Phys. Scr. T84, 18–21 (2000).
[CrossRef]

N. Efremidis, K. Hizanidis, B. A. Malomed, H. E. Nistazakis, and D. J. Frantzeskakis, “Stable transmission of solitons in the region of normal dispersion,” J. Opt. Soc. Am. B 17, 952–958 (2000).
[CrossRef]

1998 (1)

J. Atai and B. A. Malomed, “Exact stable pulses in asymmetric linearly coupled Ginzburg–Landau equations,” Phys. Lett. A 246, 412–422 (1998).
[CrossRef]

1996 (4)

B. A. Malomed and H. G. Winful, “Stable solitons in two-component active systems,” Phys. Rev. E 53, 5365–5368 (1996).
[CrossRef]

J. Atai and B. A. Malomed, “Stability and interactions of solitons in two-component active systems,” Phys. Rev. E 54, 4371–4374 (1996).
[CrossRef]

Y. Kodama, S. Wabnitz, and K. Tanaka, “Control of nonreturn-to-zero signal distortion by nonlinear gain,” Opt. Lett. 21, 719–721 (1996).
[CrossRef] [PubMed]

Y. Kodama, A. Maruta, and S. Wabnitz, “Minimum channel spacing in wavelength-division-multiplexing nonreturn-to-zero optical fiber transmissions,” Opt. Lett. 21, 1815–1817 (1996).
[CrossRef] [PubMed]

1995 (4)

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

P. A. Bélanger and N. Bélanger, “Rms characteristics of pulses in nonlinear dispersive lossy fibers,” Opt. Commun. 117, 56–60 (1995).
[CrossRef]

H. Taga, N. Edagawa, S. Yamamoto, and S. Akiba, “Recent progress in amplified undersea systems,” J. Lightwave Technol. 13, 829–834 (1995).
[CrossRef]

N. S. Bergano and C. R. Davidson, “Circulating loop transmission experiments for the study of long-haul transmission systems using erbium-doped fiber amplifiers,” J. Lightwave Technol. 13, 879–889 (1995).
[CrossRef]

1992 (2)

1991 (3)

A. Mecozzi, J. D. Moores, H. A. Haus, and Y. Lai, “Soliton transmission control,” Opt. Lett. 16, 1841–1843 (1991).
[CrossRef] [PubMed]

J. R. Simpson, H. T. Shang, L. F. Mollenauer, N. A. Olsson, P. C. Becker, K. S. Kranz, P. J. Lemaire, and M. J. Neubelt, “Performance of a distributed erbium-doped dispersion-shifted fiber amplifier,” J. Lightwave Technol. 9, 228–233 (1991).
[CrossRef]

D. Marcuse, A. R. Chraplyvy, and R. W. Tkach, “Effect of fiber nonlinearity on long-distance transmission,” J. Lightwave Technol. 9, 121–124 (1991).
[CrossRef]

1990 (1)

S. P. Craig-Ryan, B. J. Ainslie, and C. A. Millar, “Fabrication of long length of low excess loss erbium-doped optical fibre,” Electron. Lett. 26, 185–186 (1990).
[CrossRef]

1986 (1)

K. Tai, A. Tomita, and A. Hasegawa, “Observation of modulational instability in optical fibers,” Phys. Rev. Lett. 56, 135–138 (1986).
[CrossRef] [PubMed]

1984 (2)

K. Nozaki and N. Bekki, “Exact solutions of the generalized Ginzburg–Landau equation,” J. Phys. Soc. Jpn. 53, 1581–1582 (1984).
[CrossRef]

A. Hasegawa, “Generation of a train of soliton pulses by induced modulational instability in optical fibers,” Opt. Lett. 9, 288–290 (1984).
[CrossRef] [PubMed]

1983 (1)

K. Nozaki and N. Bekki, “Pattern selection and spatiotemporal transition in the Ginzburg–Landau equation,” Phys. Rev. Lett. 51, 2171–2174 (1983).
[CrossRef]

1977 (1)

N. R. Pereira and L. Stenflo, “Nonlinear Schroedinger equation including growth and damping,” Phys. Fluids 20, 1733–1734 (1977).
[CrossRef]

1972 (1)

L. M. Hocking and K. Stewartson, “On the nonlinear response of a marginally unstable plane parallel flow to a two dimensional disturbance,” Proc. R. Soc. London, Ser. A 326, 289–313 (1972).
[CrossRef]

Ainslie, B. J.

S. P. Craig-Ryan, B. J. Ainslie, and C. A. Millar, “Fabrication of long length of low excess loss erbium-doped optical fibre,” Electron. Lett. 26, 185–186 (1990).
[CrossRef]

Akiba, S.

H. Taga, N. Edagawa, S. Yamamoto, and S. Akiba, “Recent progress in amplified undersea systems,” J. Lightwave Technol. 13, 829–834 (1995).
[CrossRef]

Atai, J.

J. Atai and B. A. Malomed, “Exact stable pulses in asymmetric linearly coupled Ginzburg–Landau equations,” Phys. Lett. A 246, 412–422 (1998).
[CrossRef]

J. Atai and B. A. Malomed, “Stability and interactions of solitons in two-component active systems,” Phys. Rev. E 54, 4371–4374 (1996).
[CrossRef]

Becker, P. C.

J. R. Simpson, H. T. Shang, L. F. Mollenauer, N. A. Olsson, P. C. Becker, K. S. Kranz, P. J. Lemaire, and M. J. Neubelt, “Performance of a distributed erbium-doped dispersion-shifted fiber amplifier,” J. Lightwave Technol. 9, 228–233 (1991).
[CrossRef]

Bekki, N.

K. Nozaki and N. Bekki, “Exact solutions of the generalized Ginzburg–Landau equation,” J. Phys. Soc. Jpn. 53, 1581–1582 (1984).
[CrossRef]

K. Nozaki and N. Bekki, “Pattern selection and spatiotemporal transition in the Ginzburg–Landau equation,” Phys. Rev. Lett. 51, 2171–2174 (1983).
[CrossRef]

Bélanger, N.

P. A. Bélanger and N. Bélanger, “Rms characteristics of pulses in nonlinear dispersive lossy fibers,” Opt. Commun. 117, 56–60 (1995).
[CrossRef]

Bélanger, P. A.

P. A. Bélanger and N. Bélanger, “Rms characteristics of pulses in nonlinear dispersive lossy fibers,” Opt. Commun. 117, 56–60 (1995).
[CrossRef]

Bergano, N. S.

N. S. Bergano and C. R. Davidson, “Circulating loop transmission experiments for the study of long-haul transmission systems using erbium-doped fiber amplifiers,” J. Lightwave Technol. 13, 879–889 (1995).
[CrossRef]

Chraplyvy, A. R.

D. Marcuse, A. R. Chraplyvy, and R. W. Tkach, “Effect of fiber nonlinearity on long-distance transmission,” J. Lightwave Technol. 9, 121–124 (1991).
[CrossRef]

Craig-Ryan, S. P.

S. P. Craig-Ryan, B. J. Ainslie, and C. A. Millar, “Fabrication of long length of low excess loss erbium-doped optical fibre,” Electron. Lett. 26, 185–186 (1990).
[CrossRef]

Davidson, C. R.

N. S. Bergano and C. R. Davidson, “Circulating loop transmission experiments for the study of long-haul transmission systems using erbium-doped fiber amplifiers,” J. Lightwave Technol. 13, 879–889 (1995).
[CrossRef]

Edagawa, N.

H. Taga, N. Edagawa, S. Yamamoto, and S. Akiba, “Recent progress in amplified undersea systems,” J. Lightwave Technol. 13, 829–834 (1995).
[CrossRef]

Efremidis, N.

N. Efremidis, K. Hizanidis, B. A. Malomed, H. E. Nistazakis, and D. J. Frantzeskakis, “Stable transmission of solitons in the region of normal dispersion,” J. Opt. Soc. Am. B 17, 952–958 (2000).
[CrossRef]

N. Efremidis, K. Hizanidis, B. A. Malomed, H. E. Nistazakis, and D. J. Frantzeskakis, “Stabilizing the Pereira–Stenflo solitons in nonlinear optical fibers,” Phys. Scr. T84, 18–21 (2000).
[CrossRef]

Frantzeskakis, D. J.

N. Efremidis, K. Hizanidis, B. A. Malomed, H. E. Nistazakis, and D. J. Frantzeskakis, “Stabilizing the Pereira–Stenflo solitons in nonlinear optical fibers,” Phys. Scr. T84, 18–21 (2000).
[CrossRef]

N. Efremidis, K. Hizanidis, B. A. Malomed, H. E. Nistazakis, and D. J. Frantzeskakis, “Stable transmission of solitons in the region of normal dispersion,” J. Opt. Soc. Am. B 17, 952–958 (2000).
[CrossRef]

Hasegawa, A.

Haus, H. A.

Hizanidis, K.

N. Efremidis, K. Hizanidis, B. A. Malomed, H. E. Nistazakis, and D. J. Frantzeskakis, “Stable transmission of solitons in the region of normal dispersion,” J. Opt. Soc. Am. B 17, 952–958 (2000).
[CrossRef]

N. Efremidis, K. Hizanidis, B. A. Malomed, H. E. Nistazakis, and D. J. Frantzeskakis, “Stabilizing the Pereira–Stenflo solitons in nonlinear optical fibers,” Phys. Scr. T84, 18–21 (2000).
[CrossRef]

Hocking, L. M.

L. M. Hocking and K. Stewartson, “On the nonlinear response of a marginally unstable plane parallel flow to a two dimensional disturbance,” Proc. R. Soc. London, Ser. A 326, 289–313 (1972).
[CrossRef]

Kodama, Y.

Kranz, K. S.

J. R. Simpson, H. T. Shang, L. F. Mollenauer, N. A. Olsson, P. C. Becker, K. S. Kranz, P. J. Lemaire, and M. J. Neubelt, “Performance of a distributed erbium-doped dispersion-shifted fiber amplifier,” J. Lightwave Technol. 9, 228–233 (1991).
[CrossRef]

Lai, Y.

Lemaire, P. J.

J. R. Simpson, H. T. Shang, L. F. Mollenauer, N. A. Olsson, P. C. Becker, K. S. Kranz, P. J. Lemaire, and M. J. Neubelt, “Performance of a distributed erbium-doped dispersion-shifted fiber amplifier,” J. Lightwave Technol. 9, 228–233 (1991).
[CrossRef]

Malomed, B. A.

N. Efremidis, K. Hizanidis, B. A. Malomed, H. E. Nistazakis, and D. J. Frantzeskakis, “Stabilizing the Pereira–Stenflo solitons in nonlinear optical fibers,” Phys. Scr. T84, 18–21 (2000).
[CrossRef]

N. Efremidis, K. Hizanidis, B. A. Malomed, H. E. Nistazakis, and D. J. Frantzeskakis, “Stable transmission of solitons in the region of normal dispersion,” J. Opt. Soc. Am. B 17, 952–958 (2000).
[CrossRef]

J. Atai and B. A. Malomed, “Exact stable pulses in asymmetric linearly coupled Ginzburg–Landau equations,” Phys. Lett. A 246, 412–422 (1998).
[CrossRef]

B. A. Malomed and H. G. Winful, “Stable solitons in two-component active systems,” Phys. Rev. E 53, 5365–5368 (1996).
[CrossRef]

J. Atai and B. A. Malomed, “Stability and interactions of solitons in two-component active systems,” Phys. Rev. E 54, 4371–4374 (1996).
[CrossRef]

Marcuse, D.

D. Marcuse, “RMS width of pulses in nonlinear dispersive fibers,” J. Lightwave Technol. 10, 17–21 (1992).
[CrossRef]

D. Marcuse, A. R. Chraplyvy, and R. W. Tkach, “Effect of fiber nonlinearity on long-distance transmission,” J. Lightwave Technol. 9, 121–124 (1991).
[CrossRef]

Maruta, A.

Mecozzi, A.

Millar, C. A.

S. P. Craig-Ryan, B. J. Ainslie, and C. A. Millar, “Fabrication of long length of low excess loss erbium-doped optical fibre,” Electron. Lett. 26, 185–186 (1990).
[CrossRef]

Mollenauer, L. F.

J. R. Simpson, H. T. Shang, L. F. Mollenauer, N. A. Olsson, P. C. Becker, K. S. Kranz, P. J. Lemaire, and M. J. Neubelt, “Performance of a distributed erbium-doped dispersion-shifted fiber amplifier,” J. Lightwave Technol. 9, 228–233 (1991).
[CrossRef]

Moores, J. D.

Neubelt, M. J.

J. R. Simpson, H. T. Shang, L. F. Mollenauer, N. A. Olsson, P. C. Becker, K. S. Kranz, P. J. Lemaire, and M. J. Neubelt, “Performance of a distributed erbium-doped dispersion-shifted fiber amplifier,” J. Lightwave Technol. 9, 228–233 (1991).
[CrossRef]

Nistazakis, H. E.

N. Efremidis, K. Hizanidis, B. A. Malomed, H. E. Nistazakis, and D. J. Frantzeskakis, “Stable transmission of solitons in the region of normal dispersion,” J. Opt. Soc. Am. B 17, 952–958 (2000).
[CrossRef]

N. Efremidis, K. Hizanidis, B. A. Malomed, H. E. Nistazakis, and D. J. Frantzeskakis, “Stabilizing the Pereira–Stenflo solitons in nonlinear optical fibers,” Phys. Scr. T84, 18–21 (2000).
[CrossRef]

Nozaki, K.

K. Nozaki and N. Bekki, “Exact solutions of the generalized Ginzburg–Landau equation,” J. Phys. Soc. Jpn. 53, 1581–1582 (1984).
[CrossRef]

K. Nozaki and N. Bekki, “Pattern selection and spatiotemporal transition in the Ginzburg–Landau equation,” Phys. Rev. Lett. 51, 2171–2174 (1983).
[CrossRef]

Olsson, N. A.

J. R. Simpson, H. T. Shang, L. F. Mollenauer, N. A. Olsson, P. C. Becker, K. S. Kranz, P. J. Lemaire, and M. J. Neubelt, “Performance of a distributed erbium-doped dispersion-shifted fiber amplifier,” J. Lightwave Technol. 9, 228–233 (1991).
[CrossRef]

Pereira, N. R.

N. R. Pereira and L. Stenflo, “Nonlinear Schroedinger equation including growth and damping,” Phys. Fluids 20, 1733–1734 (1977).
[CrossRef]

Shang, H. T.

J. R. Simpson, H. T. Shang, L. F. Mollenauer, N. A. Olsson, P. C. Becker, K. S. Kranz, P. J. Lemaire, and M. J. Neubelt, “Performance of a distributed erbium-doped dispersion-shifted fiber amplifier,” J. Lightwave Technol. 9, 228–233 (1991).
[CrossRef]

Simpson, J. R.

J. R. Simpson, H. T. Shang, L. F. Mollenauer, N. A. Olsson, P. C. Becker, K. S. Kranz, P. J. Lemaire, and M. J. Neubelt, “Performance of a distributed erbium-doped dispersion-shifted fiber amplifier,” J. Lightwave Technol. 9, 228–233 (1991).
[CrossRef]

Stenflo, L.

N. R. Pereira and L. Stenflo, “Nonlinear Schroedinger equation including growth and damping,” Phys. Fluids 20, 1733–1734 (1977).
[CrossRef]

Stewartson, K.

L. M. Hocking and K. Stewartson, “On the nonlinear response of a marginally unstable plane parallel flow to a two dimensional disturbance,” Proc. R. Soc. London, Ser. A 326, 289–313 (1972).
[CrossRef]

Taga, H.

H. Taga, N. Edagawa, S. Yamamoto, and S. Akiba, “Recent progress in amplified undersea systems,” J. Lightwave Technol. 13, 829–834 (1995).
[CrossRef]

Tai, K.

K. Tai, A. Tomita, and A. Hasegawa, “Observation of modulational instability in optical fibers,” Phys. Rev. Lett. 56, 135–138 (1986).
[CrossRef] [PubMed]

Tanaka, K.

Tkach, R. W.

D. Marcuse, A. R. Chraplyvy, and R. W. Tkach, “Effect of fiber nonlinearity on long-distance transmission,” J. Lightwave Technol. 9, 121–124 (1991).
[CrossRef]

Tomita, A.

K. Tai, A. Tomita, and A. Hasegawa, “Observation of modulational instability in optical fibers,” Phys. Rev. Lett. 56, 135–138 (1986).
[CrossRef] [PubMed]

Wabnitz, S.

Winful, H. G.

B. A. Malomed and H. G. Winful, “Stable solitons in two-component active systems,” Phys. Rev. E 53, 5365–5368 (1996).
[CrossRef]

Yamamoto, S.

H. Taga, N. Edagawa, S. Yamamoto, and S. Akiba, “Recent progress in amplified undersea systems,” J. Lightwave Technol. 13, 829–834 (1995).
[CrossRef]

Electron. Lett. (1)

S. P. Craig-Ryan, B. J. Ainslie, and C. A. Millar, “Fabrication of long length of low excess loss erbium-doped optical fibre,” Electron. Lett. 26, 185–186 (1990).
[CrossRef]

J. Lightwave Technol. (5)

H. Taga, N. Edagawa, S. Yamamoto, and S. Akiba, “Recent progress in amplified undersea systems,” J. Lightwave Technol. 13, 829–834 (1995).
[CrossRef]

N. S. Bergano and C. R. Davidson, “Circulating loop transmission experiments for the study of long-haul transmission systems using erbium-doped fiber amplifiers,” J. Lightwave Technol. 13, 879–889 (1995).
[CrossRef]

D. Marcuse, A. R. Chraplyvy, and R. W. Tkach, “Effect of fiber nonlinearity on long-distance transmission,” J. Lightwave Technol. 9, 121–124 (1991).
[CrossRef]

D. Marcuse, “RMS width of pulses in nonlinear dispersive fibers,” J. Lightwave Technol. 10, 17–21 (1992).
[CrossRef]

J. R. Simpson, H. T. Shang, L. F. Mollenauer, N. A. Olsson, P. C. Becker, K. S. Kranz, P. J. Lemaire, and M. J. Neubelt, “Performance of a distributed erbium-doped dispersion-shifted fiber amplifier,” J. Lightwave Technol. 9, 228–233 (1991).
[CrossRef]

J. Opt. Soc. Am. B (1)

J. Phys. Soc. Jpn. (1)

K. Nozaki and N. Bekki, “Exact solutions of the generalized Ginzburg–Landau equation,” J. Phys. Soc. Jpn. 53, 1581–1582 (1984).
[CrossRef]

Opt. Commun. (1)

P. A. Bélanger and N. Bélanger, “Rms characteristics of pulses in nonlinear dispersive lossy fibers,” Opt. Commun. 117, 56–60 (1995).
[CrossRef]

Opt. Lett. (6)

Phys. Fluids (1)

N. R. Pereira and L. Stenflo, “Nonlinear Schroedinger equation including growth and damping,” Phys. Fluids 20, 1733–1734 (1977).
[CrossRef]

Phys. Lett. A (1)

J. Atai and B. A. Malomed, “Exact stable pulses in asymmetric linearly coupled Ginzburg–Landau equations,” Phys. Lett. A 246, 412–422 (1998).
[CrossRef]

Phys. Rev. E (2)

B. A. Malomed and H. G. Winful, “Stable solitons in two-component active systems,” Phys. Rev. E 53, 5365–5368 (1996).
[CrossRef]

J. Atai and B. A. Malomed, “Stability and interactions of solitons in two-component active systems,” Phys. Rev. E 54, 4371–4374 (1996).
[CrossRef]

Phys. Rev. Lett. (2)

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N. Efremidis, K. Hizanidis, B. A. Malomed, H. E. Nistazakis, and D. J. Frantzeskakis, “Stabilizing the Pereira–Stenflo solitons in nonlinear optical fibers,” Phys. Scr. T84, 18–21 (2000).
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Other (7)

H. Onaka, H. Miyata, G. Ishikawa, K. Otsuka, H. Ooi, Y. Kai, S. Kinoshita, M. Seino, H. Nishimoto, and T. Chikama, in Optical Amplifiers and Their Applications, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington D.C., 1996), paper PD19.

N. S. Bergano, C. R. Davidson, D. L. Wilson, F. W. Kerfoot, M. D. Trembay, M. D. Levonas, J. P. Morreale, J. D. Evankow, P. C. Corbett, M. A. Mills, G. A. Ferguson, A. M. Vengsarkar, J. R. Pedrazzani, J. A. Nagel, J. L. Zyskind, and J. W. Sulhoff, in Optical Amplifiers and Their Applications, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington D.C., 1996), paper PD23.

E. Desurvire, Erbium-Doped Fiber Amplifiers (Wiley, New York, 1994).

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

K. Hizanidis, N. Efremidis, A. Stavdas, D. J. Frantzeskakis, H. E. Nistazakis, and B. A. Malomed, “TDM and WDM with chirped solitons in optical transmission systems with distributed amplification,” in New Trends in Optical Soliton Transmission Systems, A. Hasegawa, ed. (Kluwer Academic, Dordrecht, the Netherlands, 2000), pp. 139–160.

J. R. Simpson, L. F. Mollenauer, K. S. Kranz, P. J. Lemaire, N. A. Olsson, H. T. Shang, and P. C. Becker, “A distributed erbium-doped fiber amplifier,” in Optical Fiber Communication, Vol. 1 of 1990 Technical Digest Series (Optical Society of America, Washington, D.C., 1990), paper PD19.

N. S. Bergano, C. R. Davidson, G. M. Homsey, D. J. Kalmus, P. R. Trischitta, J. Aspell, D. A. Gray, R. L. Maybach, S. Yamamoto, H. Taga, N. Edagawa, Y. Yoshida, Y. Horiuchi, T. Kawazawa, Y. Namihira, and S. Akiba, “9000 km, 5 Gb/s NRZ transmission experiment using 274 erbium-doped fiber amplifiers,” in Optical Amplifiers and Their Applications, Vol. 14 of 1994 OSA Technical Digest Series (Optical Society of America, Washington D.C., 1994), paper PD7.

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Figures (9)

Fig. 1
Fig. 1

Typical bifurcation diagram displaying the amplitude of the SW solution as a function of the coupling coefficient K for |k0|=2, Γ=4, and for various values of (a) normal- and (b) anomalous-dispersion coefficient D. (c) Detail for |D|=4, |k0|=2: The dashed (solid) curve part of the zero solution is unstable (stable) to zero-frequency perturbations. The nonzero solution is also depicted as a solid curve that consists of alternating stable and unstable parts.

Fig. 2
Fig. 2

Region of stability of the cw solution at the zero-frequency (shaded region) and the solid curve on the right separating the region of existence of SW solutions in the (Γ, K) parameter plane (a) for k0=-2, D=-4 and (b) for k0=2,D=4.

Fig. 3
Fig. 3

Instability growth rates as functions of the frequency for (a)–(c) the zero solution and for (b)–(d) the cw solution with (a), (b) k0=2, D=4, c=0.44, K=3.1 and (c)–(d) k0=-2, D=-4, c=1.08, K=2.8. The value of Γ is 2.

Fig. 4
Fig. 4

Single shock propagating laminarly ending with onset of blowup. Here, D=-4, k0=-2, K=2.8, Γ=2, and c=1.08.

Fig. 5
Fig. 5

Laminar-propagation distance as a function of the coupling coefficient K for D=4, k0=2 (crosses) and for D=-4, k0=-2 (stars).

Fig. 6
Fig. 6

Typical examples for NRZ pulse propagation: (a)–(c) anomalous-dispersion regime with D=4, k0=2, and (a) K=2.8, (c) K=3; (b)–(d) normal-dispersion regime with D=-4, k0=-2, K=2.8. (a), (b) The NRZ pulse consists of two exact shock-wave solutions; the (c)–(d) NRZ pulse consists of two initially unchirped shock-wave solutions. The value of Γ is 2.

Fig. 7
Fig. 7

(a) Laminar-propagation distance as a function of the coupling coefficient K for D=-4 and D=-8. (b) The respective bifurcation diagram for the two cases in (a) displaying the amplitude of the SW solution as a function of the coupling coefficient K. The values of the rest of the parameters are Γ=2 and k0=-2.

Fig. 8
Fig. 8

Laminar-propagation distance of an NRZ pulse as a function of the temporal pulse width, τ, for the case D=-4, k0=-2, K=2.8, and Γ=2.

Fig. 9
Fig. 9

Propagation of two pulses representing four successive bits of information (1101) for D=-4, k0=-2, K=2.8, and Γ=2. The full width at half-maximum (FWHM) of the pulses is 32 ps, and the temporal separation between them is ∼60 ps.

Equations (41)

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iuz+D2-iutt+|u|2u-iu=0,
iuz+(D/2-i)utt+|u|2u-iu-iu=Kv,
i(vz+cvt)+(k0+iΓ)v=Ku.
u=u0 exp(ikz+iθξ)(1-tanh ξ)(sech ξ)iμ,
v=v0 exp(ikz+iθξ)(1-tanh ξ)(sech ξ)iμ,
μ=-3D/4-(1/4)32+9D2,
δ3+Da1a2-k0δ2+(-K2+Γ2)δ+Γ(Γ-K2) a1a2D
-k0Γ2=0,
η2=[K2δ+k(δ2+Γ2)]Da1+(-K2Γ+δ2+Γ2)a2(δ2+Γ2)(D2a12+a22).
u02=-3µη21+D24,
c=6η1+D24.
-k-θηc+D2-i[-θ2-i(2θ+μ)]η2+u02-i
-K2-k+k0+iΓ=0,
iηc(1+iμ)+D2-i[-1-θμ+i(θ-μ)]η2-2u02
=0,
η2(2-μ2+3iμ)D2-i+u02=0.
k(0)=k01-Γ,
K(0)2=Γ1+k02(Γ-1)2.
k(2)=δa2+ΓDa1Γ-1η(1)2.
η(1)2
=-2K(0)K(2)k0(Γ+1)(δa2+ΓDa1)/(Γ-1)2+Γa2-δDa2=σK(2).
u=u1 exp[i(kz-ωt)],v=v1 exp[i(kz-ωt)].
K2-Γ(1-ω2)1+(k0+cω+Dω2/2)2(Γ-1+ω2)2>0.
{u, v}={U0, V0}exp[i(κz-ωt)],
u=(U0+U1)exp[i(κz-ωt)],
v=(V0+V1)exp[i(κz-ωt)],
-κ-ω2D2-i+2U02-IU1+U02U1*+D2-i
×(-2iωU1t+U1tt)+iU1z-KV1=0,
(-κ+k0+iΓ)V1+iV1z-KU1=0.
Q4+2i(B-Λ)Q3+(-AA-B2-Λ2+4BΛ-Δ2
-2K2)Q2+2i[K2(Λ-B)+AAΛ-BΔ2
+BΛ(B-Λ)]Q-2BK2Λ-(A+A)K2Δ
+B2(Λ2+Δ2)+AA(Λ2+Δ2)+K4=0,
A=-k-D(ω+Ω2)/2+U02+2iΩω,
A=-k-D(ω+Ω2)/2+3U02+2iΩω,
B=1-(ω2+Ω2)-iDωΩ,
Λ=Γ-icΩ,
Δ=-k+k0+cω.
u=u0 exp(ikz){exp[iθη(xi-τ/2)][1-tanh η(ξ-τ/2)][sech η(ξ-τ/2)]iμ-exp(iθξ2)[1-tanh η(ξ+τ/2)][sech η(ξ+τ/2)]iμ exp(-iϕ)},
τ=2πN+ϕη(θ+μ),
|u|2=4u021+tanh ξ22+2(1-tanh ξ)sin2μ2 lnsech ξ2eξ.

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