Abstract

The nonlinear Schrödinger equation (NLS), with its modified forms, is the central equation for the description of nonlinear pulse propagation in optical fibers. There are a number of different physical situations in which coupling between waves leads to energy transfer. In such systems, ultrashort pulses have been observed to form during propagation. In this paper we show that much of this behavior can be understood by considering the effects of gain in the NLS. We also show that perturbations of the NLS do not destroy these results, provided that the modified equation possesses solitary-wave solutions.

© 1988 Optical Society of America

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References

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  1. L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45, 1095–1098 (1980).
    [Crossref]
  2. J. Satsuma and N. Yajima, “Initial value problems of one dimensional self-modulation of nonlinear waves in dispersive media,” Prog. Theor. Suppl. 55, 284–306 (1974).
    [Crossref]
  3. L. F. Mollenauer, R. H. Stolen, J. P. Gordon, and W. J. Tomlinson, “Extreme picosecond pulse narrowing by means of soliton effect in single-mode optical fibers,” Opt. Lett. 8, 289–291 (1983).
    [Crossref] [PubMed]
  4. N. J. Doran and D. Wood, “A soliton processing element for all-optical switching and logic,” J. Opt. Soc. Am. B 4, 1843 (1987).
    [Crossref]
  5. V. A. Vysloukh and V. N. Serkin, “Generation of high energy solitons of stimulated Raman radiation in fiber light guides,” JETP Lett. 38, 199–202 (1984).
  6. E. M. Dianov, Z. S. Nikonova, A. M. Prokhorov, and V. N. Serkin, “Nonlinear dynamics of the amplification of solitons in the presence of stimulated Raman scattering in fiber-optic communication lines,” Sov. Phys. Dokl. 30, 689–691 (1986).
  7. A. S. Gouveia-Neto, A. S. L. Gomes, and J. R. Taylor, “Pulse compression, high-order solitons, and soliton-Raman generation in optical fibers, 1.3–1.45μm,” in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1987), paper MH1.
  8. J. D. Kafka and T. Baer, “Fiber Raman soliton laser pumped by a Nd:YAG laser,” Opt. Lett. 12, 181–183 (1987).
    [Crossref] [PubMed]
  9. E. M. Dianov, A. Ya. Karasik, P. V. Mamyshev, A. M. Prokhorov, V. N. Serkin, M. F. Stel’makh, and A. A. Fomichev, “Stimulated Raman conversion of multisoliton pulses in quartz optical fibers,” Pisma Zh. Eksp. Teor. Fiz. 41, 242–244 (1985).
  10. K. J. Blow, N. J. Doran, and D. Wood, “Polarization instabilities for solitons in birefringent fibers,” Opt. Lett. 12, 202–204 (1987).
    [Crossref] [PubMed]
  11. K. J. Blow and N. J. Doran, “The asymptotic dispersion of soliton pulses in lossy fibres,” Opt. Commun. 52, 367–370 (1985).
    [Crossref]
  12. F. M. Mitschke and L. F. Mollenauer, “Discovery of the soliton self-frequency shift,” Opt. Lett. 11, 659–661 (1986).
    [Crossref] [PubMed]
  13. J. P. Gordon, “Theory of the soliton self-frequency shift,” Opt. Lett. 11, 662–664 (1986).
    [Crossref] [PubMed]
  14. B. P. Nelson, D. Cotter, K. J. Blow, and N. J. Doran, “Large nonlinear pulse broadening in long lengths of monomode fibre,” Opt. Commun. 48, 292–294 (1983).
    [Crossref]
  15. A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. 1. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
    [Crossref]
  16. A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. 2. Normal dispersion,” Appl. Phys. Lett. 23, 171–172 (1973).
    [Crossref]
  17. N. J. Doran and K. J. Blow, “Solitons in optical communications,” IEEE J. Quantum Electron. QE-19, 1883–1888 (1983).
    [Crossref]
  18. D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 27, 3135–3145 (1983).
    [Crossref]
  19. K. J. Blow, N. J. Doran, and D. Wood, “Trapping of energy into solitary waves in amplified nonlinear dispersive systems,” Opt. Lett. 12, 1011–1013 (1987).
    [Crossref] [PubMed]
  20. G. R. Lamb, Elements of Soliton Theory (Wiley Interscience, New York, 1980).
  21. V. E. Zakharov and A. B. Shabat, “Exact theory of two dimensional self focusing and one dimensional self modulation of nonlinear waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).
  22. K. J. Blow, N. J. Doran, and E. Cummins, “Nonlinear limits on bandwidth at the minimum dispersion in optical fibres”, Opt. Commun. 48, 181–184 (1983).
    [Crossref]
  23. P. L. Chu and C. Desem, “Effect of third order dispersion of optical fibre on soliton interaction,” Electron. Lett. 21, 228–229 (1985).
    [Crossref]
  24. V. A. Vysloukh, “Propagation of pulses in optical fibers in the region of a dispersion minimum role of nonlinearity and higher-order dispersion,” Sov. J. Quantum Electron. 13, 1113–1114 (1983).
    [Crossref]
  25. A. Hasegawa and Y. Kodama, “Signal transmission by optical solitons in monomode fiber,” Proc. IEEE 69, 1145–1150 (1981).
    [Crossref]
  26. P. K. A. Wai, C. R. Menyuk, Y. C. Lee, and H. H. Chen, Nonlinear pulse propagation in the neighborhood of the zero-dispersion wavelength of monomode optical fibers,” Opt. Lett. 11, 464–466 (1986).
    [Crossref] [PubMed]
  27. P. K. A. Wai, C. R. Menyuk, H. H. Chen, and Y. C. Lee, “Soliton at the zero group dispersion wavelength of a single mode fiber,” Opt. Lett. 12, 628–630 (1987).
    [Crossref] [PubMed]
  28. C. S. Bretherton and E. A. Spiegel, “Intermittency through modulational instability,” Phys. Lett. A 96, 152–156 (1983).
    [Crossref]

1987 (5)

1986 (4)

1985 (3)

E. M. Dianov, A. Ya. Karasik, P. V. Mamyshev, A. M. Prokhorov, V. N. Serkin, M. F. Stel’makh, and A. A. Fomichev, “Stimulated Raman conversion of multisoliton pulses in quartz optical fibers,” Pisma Zh. Eksp. Teor. Fiz. 41, 242–244 (1985).

K. J. Blow and N. J. Doran, “The asymptotic dispersion of soliton pulses in lossy fibres,” Opt. Commun. 52, 367–370 (1985).
[Crossref]

P. L. Chu and C. Desem, “Effect of third order dispersion of optical fibre on soliton interaction,” Electron. Lett. 21, 228–229 (1985).
[Crossref]

1984 (1)

V. A. Vysloukh and V. N. Serkin, “Generation of high energy solitons of stimulated Raman radiation in fiber light guides,” JETP Lett. 38, 199–202 (1984).

1983 (7)

L. F. Mollenauer, R. H. Stolen, J. P. Gordon, and W. J. Tomlinson, “Extreme picosecond pulse narrowing by means of soliton effect in single-mode optical fibers,” Opt. Lett. 8, 289–291 (1983).
[Crossref] [PubMed]

N. J. Doran and K. J. Blow, “Solitons in optical communications,” IEEE J. Quantum Electron. QE-19, 1883–1888 (1983).
[Crossref]

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

B. P. Nelson, D. Cotter, K. J. Blow, and N. J. Doran, “Large nonlinear pulse broadening in long lengths of monomode fibre,” Opt. Commun. 48, 292–294 (1983).
[Crossref]

V. A. Vysloukh, “Propagation of pulses in optical fibers in the region of a dispersion minimum role of nonlinearity and higher-order dispersion,” Sov. J. Quantum Electron. 13, 1113–1114 (1983).
[Crossref]

K. J. Blow, N. J. Doran, and E. Cummins, “Nonlinear limits on bandwidth at the minimum dispersion in optical fibres”, Opt. Commun. 48, 181–184 (1983).
[Crossref]

C. S. Bretherton and E. A. Spiegel, “Intermittency through modulational instability,” Phys. Lett. A 96, 152–156 (1983).
[Crossref]

1981 (1)

A. Hasegawa and Y. Kodama, “Signal transmission by optical solitons in monomode fiber,” Proc. IEEE 69, 1145–1150 (1981).
[Crossref]

1980 (1)

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45, 1095–1098 (1980).
[Crossref]

1974 (1)

J. Satsuma and N. Yajima, “Initial value problems of one dimensional self-modulation of nonlinear waves in dispersive media,” Prog. Theor. Suppl. 55, 284–306 (1974).
[Crossref]

1973 (2)

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. 1. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
[Crossref]

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. 2. Normal dispersion,” Appl. Phys. Lett. 23, 171–172 (1973).
[Crossref]

1972 (1)

V. E. Zakharov and A. B. Shabat, “Exact theory of two dimensional self focusing and one dimensional self modulation of nonlinear waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

Anderson, D.

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

Baer, T.

Blow, K. J.

K. J. Blow, N. J. Doran, and D. Wood, “Polarization instabilities for solitons in birefringent fibers,” Opt. Lett. 12, 202–204 (1987).
[Crossref] [PubMed]

K. J. Blow, N. J. Doran, and D. Wood, “Trapping of energy into solitary waves in amplified nonlinear dispersive systems,” Opt. Lett. 12, 1011–1013 (1987).
[Crossref] [PubMed]

K. J. Blow and N. J. Doran, “The asymptotic dispersion of soliton pulses in lossy fibres,” Opt. Commun. 52, 367–370 (1985).
[Crossref]

K. J. Blow, N. J. Doran, and E. Cummins, “Nonlinear limits on bandwidth at the minimum dispersion in optical fibres”, Opt. Commun. 48, 181–184 (1983).
[Crossref]

N. J. Doran and K. J. Blow, “Solitons in optical communications,” IEEE J. Quantum Electron. QE-19, 1883–1888 (1983).
[Crossref]

B. P. Nelson, D. Cotter, K. J. Blow, and N. J. Doran, “Large nonlinear pulse broadening in long lengths of monomode fibre,” Opt. Commun. 48, 292–294 (1983).
[Crossref]

Bretherton, C. S.

C. S. Bretherton and E. A. Spiegel, “Intermittency through modulational instability,” Phys. Lett. A 96, 152–156 (1983).
[Crossref]

Chen, H. H.

Chu, P. L.

P. L. Chu and C. Desem, “Effect of third order dispersion of optical fibre on soliton interaction,” Electron. Lett. 21, 228–229 (1985).
[Crossref]

Cotter, D.

B. P. Nelson, D. Cotter, K. J. Blow, and N. J. Doran, “Large nonlinear pulse broadening in long lengths of monomode fibre,” Opt. Commun. 48, 292–294 (1983).
[Crossref]

Cummins, E.

K. J. Blow, N. J. Doran, and E. Cummins, “Nonlinear limits on bandwidth at the minimum dispersion in optical fibres”, Opt. Commun. 48, 181–184 (1983).
[Crossref]

Desem, C.

P. L. Chu and C. Desem, “Effect of third order dispersion of optical fibre on soliton interaction,” Electron. Lett. 21, 228–229 (1985).
[Crossref]

Dianov, E. M.

E. M. Dianov, Z. S. Nikonova, A. M. Prokhorov, and V. N. Serkin, “Nonlinear dynamics of the amplification of solitons in the presence of stimulated Raman scattering in fiber-optic communication lines,” Sov. Phys. Dokl. 30, 689–691 (1986).

E. M. Dianov, A. Ya. Karasik, P. V. Mamyshev, A. M. Prokhorov, V. N. Serkin, M. F. Stel’makh, and A. A. Fomichev, “Stimulated Raman conversion of multisoliton pulses in quartz optical fibers,” Pisma Zh. Eksp. Teor. Fiz. 41, 242–244 (1985).

Doran, N. J.

N. J. Doran and D. Wood, “A soliton processing element for all-optical switching and logic,” J. Opt. Soc. Am. B 4, 1843 (1987).
[Crossref]

K. J. Blow, N. J. Doran, and D. Wood, “Polarization instabilities for solitons in birefringent fibers,” Opt. Lett. 12, 202–204 (1987).
[Crossref] [PubMed]

K. J. Blow, N. J. Doran, and D. Wood, “Trapping of energy into solitary waves in amplified nonlinear dispersive systems,” Opt. Lett. 12, 1011–1013 (1987).
[Crossref] [PubMed]

K. J. Blow and N. J. Doran, “The asymptotic dispersion of soliton pulses in lossy fibres,” Opt. Commun. 52, 367–370 (1985).
[Crossref]

B. P. Nelson, D. Cotter, K. J. Blow, and N. J. Doran, “Large nonlinear pulse broadening in long lengths of monomode fibre,” Opt. Commun. 48, 292–294 (1983).
[Crossref]

N. J. Doran and K. J. Blow, “Solitons in optical communications,” IEEE J. Quantum Electron. QE-19, 1883–1888 (1983).
[Crossref]

K. J. Blow, N. J. Doran, and E. Cummins, “Nonlinear limits on bandwidth at the minimum dispersion in optical fibres”, Opt. Commun. 48, 181–184 (1983).
[Crossref]

Fomichev, A. A.

E. M. Dianov, A. Ya. Karasik, P. V. Mamyshev, A. M. Prokhorov, V. N. Serkin, M. F. Stel’makh, and A. A. Fomichev, “Stimulated Raman conversion of multisoliton pulses in quartz optical fibers,” Pisma Zh. Eksp. Teor. Fiz. 41, 242–244 (1985).

Gomes, A. S. L.

A. S. Gouveia-Neto, A. S. L. Gomes, and J. R. Taylor, “Pulse compression, high-order solitons, and soliton-Raman generation in optical fibers, 1.3–1.45μm,” in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1987), paper MH1.

Gordon, J. P.

Gouveia-Neto, A. S.

A. S. Gouveia-Neto, A. S. L. Gomes, and J. R. Taylor, “Pulse compression, high-order solitons, and soliton-Raman generation in optical fibers, 1.3–1.45μm,” in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1987), paper MH1.

Hasegawa, A.

A. Hasegawa and Y. Kodama, “Signal transmission by optical solitons in monomode fiber,” Proc. IEEE 69, 1145–1150 (1981).
[Crossref]

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. 1. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
[Crossref]

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. 2. Normal dispersion,” Appl. Phys. Lett. 23, 171–172 (1973).
[Crossref]

Kafka, J. D.

Karasik, A. Ya.

E. M. Dianov, A. Ya. Karasik, P. V. Mamyshev, A. M. Prokhorov, V. N. Serkin, M. F. Stel’makh, and A. A. Fomichev, “Stimulated Raman conversion of multisoliton pulses in quartz optical fibers,” Pisma Zh. Eksp. Teor. Fiz. 41, 242–244 (1985).

Kodama, Y.

A. Hasegawa and Y. Kodama, “Signal transmission by optical solitons in monomode fiber,” Proc. IEEE 69, 1145–1150 (1981).
[Crossref]

Lamb, G. R.

G. R. Lamb, Elements of Soliton Theory (Wiley Interscience, New York, 1980).

Lee, Y. C.

Mamyshev, P. V.

E. M. Dianov, A. Ya. Karasik, P. V. Mamyshev, A. M. Prokhorov, V. N. Serkin, M. F. Stel’makh, and A. A. Fomichev, “Stimulated Raman conversion of multisoliton pulses in quartz optical fibers,” Pisma Zh. Eksp. Teor. Fiz. 41, 242–244 (1985).

Menyuk, C. R.

Mitschke, F. M.

Mollenauer, L. F.

Nelson, B. P.

B. P. Nelson, D. Cotter, K. J. Blow, and N. J. Doran, “Large nonlinear pulse broadening in long lengths of monomode fibre,” Opt. Commun. 48, 292–294 (1983).
[Crossref]

Nikonova, Z. S.

E. M. Dianov, Z. S. Nikonova, A. M. Prokhorov, and V. N. Serkin, “Nonlinear dynamics of the amplification of solitons in the presence of stimulated Raman scattering in fiber-optic communication lines,” Sov. Phys. Dokl. 30, 689–691 (1986).

Prokhorov, A. M.

E. M. Dianov, Z. S. Nikonova, A. M. Prokhorov, and V. N. Serkin, “Nonlinear dynamics of the amplification of solitons in the presence of stimulated Raman scattering in fiber-optic communication lines,” Sov. Phys. Dokl. 30, 689–691 (1986).

E. M. Dianov, A. Ya. Karasik, P. V. Mamyshev, A. M. Prokhorov, V. N. Serkin, M. F. Stel’makh, and A. A. Fomichev, “Stimulated Raman conversion of multisoliton pulses in quartz optical fibers,” Pisma Zh. Eksp. Teor. Fiz. 41, 242–244 (1985).

Satsuma, J.

J. Satsuma and N. Yajima, “Initial value problems of one dimensional self-modulation of nonlinear waves in dispersive media,” Prog. Theor. Suppl. 55, 284–306 (1974).
[Crossref]

Serkin, V. N.

E. M. Dianov, Z. S. Nikonova, A. M. Prokhorov, and V. N. Serkin, “Nonlinear dynamics of the amplification of solitons in the presence of stimulated Raman scattering in fiber-optic communication lines,” Sov. Phys. Dokl. 30, 689–691 (1986).

E. M. Dianov, A. Ya. Karasik, P. V. Mamyshev, A. M. Prokhorov, V. N. Serkin, M. F. Stel’makh, and A. A. Fomichev, “Stimulated Raman conversion of multisoliton pulses in quartz optical fibers,” Pisma Zh. Eksp. Teor. Fiz. 41, 242–244 (1985).

V. A. Vysloukh and V. N. Serkin, “Generation of high energy solitons of stimulated Raman radiation in fiber light guides,” JETP Lett. 38, 199–202 (1984).

Shabat, A. B.

V. E. Zakharov and A. B. Shabat, “Exact theory of two dimensional self focusing and one dimensional self modulation of nonlinear waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

Spiegel, E. A.

C. S. Bretherton and E. A. Spiegel, “Intermittency through modulational instability,” Phys. Lett. A 96, 152–156 (1983).
[Crossref]

Stel’makh, M. F.

E. M. Dianov, A. Ya. Karasik, P. V. Mamyshev, A. M. Prokhorov, V. N. Serkin, M. F. Stel’makh, and A. A. Fomichev, “Stimulated Raman conversion of multisoliton pulses in quartz optical fibers,” Pisma Zh. Eksp. Teor. Fiz. 41, 242–244 (1985).

Stolen, R. H.

L. F. Mollenauer, R. H. Stolen, J. P. Gordon, and W. J. Tomlinson, “Extreme picosecond pulse narrowing by means of soliton effect in single-mode optical fibers,” Opt. Lett. 8, 289–291 (1983).
[Crossref] [PubMed]

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45, 1095–1098 (1980).
[Crossref]

Tappert, F.

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. 1. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
[Crossref]

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. 2. Normal dispersion,” Appl. Phys. Lett. 23, 171–172 (1973).
[Crossref]

Taylor, J. R.

A. S. Gouveia-Neto, A. S. L. Gomes, and J. R. Taylor, “Pulse compression, high-order solitons, and soliton-Raman generation in optical fibers, 1.3–1.45μm,” in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1987), paper MH1.

Tomlinson, W. J.

Vysloukh, V. A.

V. A. Vysloukh and V. N. Serkin, “Generation of high energy solitons of stimulated Raman radiation in fiber light guides,” JETP Lett. 38, 199–202 (1984).

V. A. Vysloukh, “Propagation of pulses in optical fibers in the region of a dispersion minimum role of nonlinearity and higher-order dispersion,” Sov. J. Quantum Electron. 13, 1113–1114 (1983).
[Crossref]

Wai, P. K. A.

Wood, D.

Yajima, N.

J. Satsuma and N. Yajima, “Initial value problems of one dimensional self-modulation of nonlinear waves in dispersive media,” Prog. Theor. Suppl. 55, 284–306 (1974).
[Crossref]

Zakharov, V. E.

V. E. Zakharov and A. B. Shabat, “Exact theory of two dimensional self focusing and one dimensional self modulation of nonlinear waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

Appl. Phys. Lett. (2)

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. 1. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
[Crossref]

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. 2. Normal dispersion,” Appl. Phys. Lett. 23, 171–172 (1973).
[Crossref]

Electron. Lett. (1)

P. L. Chu and C. Desem, “Effect of third order dispersion of optical fibre on soliton interaction,” Electron. Lett. 21, 228–229 (1985).
[Crossref]

IEEE J. Quantum Electron. (1)

N. J. Doran and K. J. Blow, “Solitons in optical communications,” IEEE J. Quantum Electron. QE-19, 1883–1888 (1983).
[Crossref]

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

JETP Lett. (1)

V. A. Vysloukh and V. N. Serkin, “Generation of high energy solitons of stimulated Raman radiation in fiber light guides,” JETP Lett. 38, 199–202 (1984).

Opt. Commun. (3)

K. J. Blow and N. J. Doran, “The asymptotic dispersion of soliton pulses in lossy fibres,” Opt. Commun. 52, 367–370 (1985).
[Crossref]

K. J. Blow, N. J. Doran, and E. Cummins, “Nonlinear limits on bandwidth at the minimum dispersion in optical fibres”, Opt. Commun. 48, 181–184 (1983).
[Crossref]

B. P. Nelson, D. Cotter, K. J. Blow, and N. J. Doran, “Large nonlinear pulse broadening in long lengths of monomode fibre,” Opt. Commun. 48, 292–294 (1983).
[Crossref]

Opt. Lett. (8)

Phys. Lett. A (1)

C. S. Bretherton and E. A. Spiegel, “Intermittency through modulational instability,” Phys. Lett. A 96, 152–156 (1983).
[Crossref]

Phys. Rev. A (1)

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

Phys. Rev. Lett. (1)

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45, 1095–1098 (1980).
[Crossref]

Pisma Zh. Eksp. Teor. Fiz. (1)

E. M. Dianov, A. Ya. Karasik, P. V. Mamyshev, A. M. Prokhorov, V. N. Serkin, M. F. Stel’makh, and A. A. Fomichev, “Stimulated Raman conversion of multisoliton pulses in quartz optical fibers,” Pisma Zh. Eksp. Teor. Fiz. 41, 242–244 (1985).

Proc. IEEE (1)

A. Hasegawa and Y. Kodama, “Signal transmission by optical solitons in monomode fiber,” Proc. IEEE 69, 1145–1150 (1981).
[Crossref]

Prog. Theor. Suppl. (1)

J. Satsuma and N. Yajima, “Initial value problems of one dimensional self-modulation of nonlinear waves in dispersive media,” Prog. Theor. Suppl. 55, 284–306 (1974).
[Crossref]

Sov. J. Quantum Electron. (1)

V. A. Vysloukh, “Propagation of pulses in optical fibers in the region of a dispersion minimum role of nonlinearity and higher-order dispersion,” Sov. J. Quantum Electron. 13, 1113–1114 (1983).
[Crossref]

Sov. Phys. Dokl. (1)

E. M. Dianov, Z. S. Nikonova, A. M. Prokhorov, and V. N. Serkin, “Nonlinear dynamics of the amplification of solitons in the presence of stimulated Raman scattering in fiber-optic communication lines,” Sov. Phys. Dokl. 30, 689–691 (1986).

Sov. Phys. JETP (1)

V. E. Zakharov and A. B. Shabat, “Exact theory of two dimensional self focusing and one dimensional self modulation of nonlinear waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

Other (2)

G. R. Lamb, Elements of Soliton Theory (Wiley Interscience, New York, 1980).

A. S. Gouveia-Neto, A. S. L. Gomes, and J. R. Taylor, “Pulse compression, high-order solitons, and soliton-Raman generation in optical fibers, 1.3–1.45μm,” in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1987), paper MH1.

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Figures (13)

Fig. 1
Fig. 1

Evolution of solitons in the presence of gain (Γ = 0.2): (a) N = 1, (b) N = 2. The periodic boundaries are at t = ±5. In all the evolution plots presented here the vertical axis is the pulse intensity.

Fig. 2
Fig. 2

Evolution of the pulse width, W, for an initial single soliton. (a) Γ = 0.2 with z > 0 for gain and z < 0 for loss, (b) Γ = 1.

Fig. 3
Fig. 3

Figure 2(a) replotted for positive z with a rescaled pulse width, W′ = ezW. Note the decrease in the amplitude of the oscillations during the evolution.

Fig. 4
Fig. 4

Soliton energy, E, normalized to the total energy, as obtained from the Zakharov–Shabat eigenvalues for an initial single soliton.

Fig. 5
Fig. 5

Evolution of bandwidth-limited noise with Γ = 0.2. A single soliton eventually dominates and becomes the asymptotic state.

Fig. 6
Fig. 6

Evolution of pulses in Eq. (15) for α3 = 0, α4 = 0.0005, and Γ = 0.2: (a) N = 1, (b) N = 4. The periodic boundaries are at t = ±5.

Fig. 7
Fig. 7

As in Fig. 6(b) with the gain turned off after z = 2.5. The final pulses are nondispersive, which is indicative of their solitary-wave nature.

Fig. 8
Fig. 8

Evolution of the pulse width, W, for the single soliton shown in Fig. 6(a).

Fig. 9
Fig. 9

Evolution as in Fig. 6(a) but with α4 = −0.0005.

Fig. 10
Fig. 10

Evolution of pulses in Eq. (15) for α3 = 0.001, α4 = 0, and Γ = 0.2: (a) N = 4, (b) starting from noise. The periodic boundaries are at t = ±5.

Fig. 11
Fig. 11

Evolution with a time-delayed nonlinearity to model intrapulse Raman scatterings: (a) N = 1, (b) N = 4. The periodic boundaries are at t = ±5.

Fig. 12
Fig. 12

Evolution of a single soliton with bandwidth-limited gain, μ = 0.01.

Fig. 13
Fig. 13

Comparison of the pulse width, W, obtained numerically from the calculation used for Fig. 12 with the prediction of Eq. (20). Also shown is the comparison for the same problem but withy μ = 0.

Equations (22)

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i u z + 1 2 2 u t 2 + u 2 u = 0.
i ( U p z - δ 2 U p t ) + β p 2 U p t 2 + U p 2 U p = - ( i + 2 ) U s 2 U p ,
i ( U s z + δ 2 U s t ) + β s 2 U s t 2 + U s 2 U s = ( i - 2 ) U p 2 U s ,
i ( A z + δ 2 A t ) + β 2 A t 2 + A 2 A = - κ B - 2 B 2 A ,
i ( B z - δ 2 B t ) + β 2 B t 2 + B 2 B = - κ A - 2 A 2 B ,
i u z + 1 2 2 u t 2 + u 2 u = i Γ u ,
t = t e 2 Γ z , z = e 4 Γ z - 1 4 Γ .
d β d z = 2 Γ β ,
exp [ - i ( 1 - e 4 Γ z ) 8 Γ ] e 2 Γ z sech ( t e 2 Γ z ) .
v = u e - 2 Γ z ,
i v z + 1 2 2 v t 2 + v 2 v = i Γ ( 4 Γ z + 1 ) ( v + 2 t v t ) .
N 2 exp ( 2 Γ c z ) = ( N + n ) 2 .
N 2 exp [ Γ c π ( N + n ) 2 ] ~ ( N + n ) 2 .
Γ c ~ 2 π ( N + n ) 2 ln ( N + n N ) .
Γ c ~ 2 π N n ,
Γ c ~ 2 π n 2 ln n ,
i u z + 1 2 2 u t 2 + u 2 u = i α 3 3 u t 3 + α 4 4 u t 4 + i Γ u .
u f ( t - τ ) u ( τ ) 2 d τ
Γ ( ω ) = Γ 0 - μ ω 2 .
i u z + 1 / 2 2 u t 2 + u 2 u = i Γ 0 u + i μ 2 u t 2 .
d β d z = 2 ( Γ 0 β - μ 3 β 3 ) ,
β = ( 3 Γ 0 μ ) 1 / 2 [ 1 + ( 3 Γ 0 μ - 1 ) exp ( - 4 Γ 0 z ) ] - 1 / 2 .

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