Abstract

We examine the stationary propagation of a black solitary wave in a fiber laser or in a fiber transmission system with periodically distributed amplifiers and saturable absorbers that is governed by the Ginzburg–Landau equation. An analytical solution of the chirped black solitary wave to the Ginzburg–Landau equation that includes the nonlinear saturation effect is obtained for what we believe to be the first time. The stability analyses reveal that the stationary propagation of the chirped black solitary wave can be stable when the saturation effect of nonlinear gain or loss is taken into account, whereas the chirped black solitary-wave solution of the Ginzburg–Landau equation that does not include the nonlinear saturation of gain or loss is found to be unstable. The criterion for the stable or unstable propagation of the chirped black solitary wave in the presence of the nonlinear gain or loss saturation is presented. Also, it is shown that two identical chirped black solitary waves launched in parallel will attract each other and may develop into a bound state of two parallel chirped black solitary waves. This is in contrast to the behavior of conventional black solitons of an unperturbed system, in which the two black solitons launched in parallel repel each other and distance themselves during propagation.

© 1996 Optical Society of America

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Corrections

Yijiang Chen, "Stable chirped black solitary waves in dispersive media with intensity-dependent gain and loss: erratum," Opt. Lett. 21, 1009-1009 (1996)
https://www.osapublishing.org/ol/abstract.cfm?uri=ol-21-13-1009

References

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]

1995 (1)

1994 (1)

1993 (3)

C. R. Menyuk, J. Opt. Soc. Am. B 10, 1585 (1993).
[CrossRef]

L. F. Mollenauer, E. Lichtman, M. J. Neubelt, G. T. Harvey, Electron. Lett. 29, 910 (1993).
[CrossRef]

J. D. Moores, Opt. Commun. 96, 65 (1993).
[CrossRef]

1992 (1)

Y. Kodama, M. Romagnoli, S. Wabnitz, Electron. Lett. 28, 1981 (1992).

1991 (4)

1989 (3)

1977 (1)

N. R. Pereira, L. Stenflo, Phys. Fluids 20, 1733 (1977).
[CrossRef]

1973 (2)

V. E. Zakharov, A. B. Shabat, Sov. Phys. JETP 37, 823 (1973).

A. Hasegawa, F. Tappert, Appl. Phys. Lett. 23, 142, 171 (1973).
[CrossRef]

1922 (1)

Y. Chen, Phys. Rev. A 45, 6922 (1922).
[CrossRef]

Afanasjev, V. V.

Atai, J.

Belanger, P. A.

P. A. Belanger, L. Gagnon, C. Pare, Opt. Lett. 14, 943 (1989).
[CrossRef] [PubMed]

C. Pare, L. Gagnon, P. A. Belanger, Opt. Commun. 74, 228 (1989).
[CrossRef]

Bourkoff, E.

Chen, C. J.

Chen, Y.

Y. Chen, Electron. Lett. 27, 1985 (1991).
[CrossRef]

Y. Chen, J. Atai, Opt. Lett. 16, 1933 (1991).

Y. Chen, Phys. Rev. A 45, 6922 (1922).
[CrossRef]

Faddev, L. D.

L. D. Faddev, L. A. Takhtajan, Hamiltonaian Methods in the Theory of Solitons (Springer-Verlag, Berlin, 1987); p. 19I. M. Uzunov, V. S. Gerdjikov, Phys. Rev. E. 47, 1589 (1993)Y. S. Kivshar, X. Yang, Phys. Rev. E 49, 1657 (1994)H. Ikeda, M. Matsumoto, A. Hasegawa, Opt. Lett. 20, 1113 (1995).
[CrossRef] [PubMed]

Fujimoto, J. G.

Gagnon, L.

C. Pare, L. Gagnon, P. A. Belanger, Opt. Commun. 74, 228 (1989).
[CrossRef]

P. A. Belanger, L. Gagnon, C. Pare, Opt. Lett. 14, 943 (1989).
[CrossRef] [PubMed]

Harvey, G. T.

L. F. Mollenauer, E. Lichtman, M. J. Neubelt, G. T. Harvey, Electron. Lett. 29, 910 (1993).
[CrossRef]

Hasegawa, A.

A. Hasegawa, Y. Kodama, Phys. Rev. Lett. 66, 161 (1991).
[CrossRef] [PubMed]

A. Hasegawa, F. Tappert, Appl. Phys. Lett. 23, 142, 171 (1973).
[CrossRef]

Haus, H. A.

Ippen, E. P.

Kodama, Y.

Y. Kodama, M. Romagnoli, S. Wabnitz, Electron. Lett. 28, 1981 (1992).

A. Hasegawa, Y. Kodama, Phys. Rev. Lett. 66, 161 (1991).
[CrossRef] [PubMed]

Lichtman, E.

L. F. Mollenauer, E. Lichtman, M. J. Neubelt, G. T. Harvey, Electron. Lett. 29, 910 (1993).
[CrossRef]

Menyuk, C. R.

Mollenauer, L. F.

L. F. Mollenauer, E. Lichtman, M. J. Neubelt, G. T. Harvey, Electron. Lett. 29, 910 (1993).
[CrossRef]

Moores, J. D.

J. D. Moores, Opt. Commun. 96, 65 (1993).
[CrossRef]

Neubelt, M. J.

L. F. Mollenauer, E. Lichtman, M. J. Neubelt, G. T. Harvey, Electron. Lett. 29, 910 (1993).
[CrossRef]

Pare, C.

C. Pare, L. Gagnon, P. A. Belanger, Opt. Commun. 74, 228 (1989).
[CrossRef]

P. A. Belanger, L. Gagnon, C. Pare, Opt. Lett. 14, 943 (1989).
[CrossRef] [PubMed]

Pereira, N. R.

N. R. Pereira, L. Stenflo, Phys. Fluids 20, 1733 (1977).
[CrossRef]

Romagnoli, M.

Y. Kodama, M. Romagnoli, S. Wabnitz, Electron. Lett. 28, 1981 (1992).

Shabat, A. B.

V. E. Zakharov, A. B. Shabat, Sov. Phys. JETP 37, 823 (1973).

Stenflo, L.

N. R. Pereira, L. Stenflo, Phys. Fluids 20, 1733 (1977).
[CrossRef]

Takhtajan, L. A.

L. D. Faddev, L. A. Takhtajan, Hamiltonaian Methods in the Theory of Solitons (Springer-Verlag, Berlin, 1987); p. 19I. M. Uzunov, V. S. Gerdjikov, Phys. Rev. E. 47, 1589 (1993)Y. S. Kivshar, X. Yang, Phys. Rev. E 49, 1657 (1994)H. Ikeda, M. Matsumoto, A. Hasegawa, Opt. Lett. 20, 1113 (1995).
[CrossRef] [PubMed]

Tappert, F.

A. Hasegawa, F. Tappert, Appl. Phys. Lett. 23, 142, 171 (1973).
[CrossRef]

Wabnitz, S.

Y. Kodama, M. Romagnoli, S. Wabnitz, Electron. Lett. 28, 1981 (1992).

Wai, P. K. A.

Zakharov, V. E.

V. E. Zakharov, A. B. Shabat, Sov. Phys. JETP 37, 823 (1973).

Zhao, W.

Appl. Phys. Lett. (1)

A. Hasegawa, F. Tappert, Appl. Phys. Lett. 23, 142, 171 (1973).
[CrossRef]

Electron. Lett. (3)

L. F. Mollenauer, E. Lichtman, M. J. Neubelt, G. T. Harvey, Electron. Lett. 29, 910 (1993).
[CrossRef]

Y. Kodama, M. Romagnoli, S. Wabnitz, Electron. Lett. 28, 1981 (1992).

Y. Chen, Electron. Lett. 27, 1985 (1991).
[CrossRef]

J. Opt. Soc. Am. B (2)

Opt. Commun. (2)

J. D. Moores, Opt. Commun. 96, 65 (1993).
[CrossRef]

C. Pare, L. Gagnon, P. A. Belanger, Opt. Commun. 74, 228 (1989).
[CrossRef]

Opt. Lett. (5)

Phys. Fluids (1)

N. R. Pereira, L. Stenflo, Phys. Fluids 20, 1733 (1977).
[CrossRef]

Phys. Rev. A (1)

Y. Chen, Phys. Rev. A 45, 6922 (1922).
[CrossRef]

Phys. Rev. Lett. (1)

A. Hasegawa, Y. Kodama, Phys. Rev. Lett. 66, 161 (1991).
[CrossRef] [PubMed]

Sov. Phys. JETP (1)

V. E. Zakharov, A. B. Shabat, Sov. Phys. JETP 37, 823 (1973).

Other (1)

L. D. Faddev, L. A. Takhtajan, Hamiltonaian Methods in the Theory of Solitons (Springer-Verlag, Berlin, 1987); p. 19I. M. Uzunov, V. S. Gerdjikov, Phys. Rev. E. 47, 1589 (1993)Y. S. Kivshar, X. Yang, Phys. Rev. E 49, 1657 (1994)H. Ikeda, M. Matsumoto, A. Hasegawa, Opt. Lett. 20, 1113 (1995).
[CrossRef] [PubMed]

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Figures (3)

Fig. 1
Fig. 1

Demonstration of the propagaton of the chirped black solitary-wave solution of Eq. (2) for α2 = 0.2 of the two-photon absorption, α = 0.1 of the net constant gain, and γ2 = 0.2, showing (a) stability to an initially imposed symmetric perturbation and (b) instability to an asymmetric perturbation.

Fig. 2
Fig. 2

Demonstration of (a) the stable evolution of the chirped solitary-wave solution of Eq. (3) for α2 = −0.2, α = −0.0985, α4 = −0.1, and γ2 = 0 and (b) the unstable evolution of the chirped solitary-wave solution of Eq. (3) for α2 = 0.2, α = 0.0985, α4 = 0.1, and γ2 = 0.

Fig. 3
Fig. 3

Demonstration of (a) the repulsion of the two conventional black solitons launched in parallel in the unperturbed system of α2 = α = α4 = γ2 = 0 and (b) the attraction of the two chirped black solitary waves of Eq. (3) launched in parallel (at α2 = −0.2, α = −0.0985, α4 = −0.1, and γ2 = 0) and eventually evolving to a bound state of the two parallel chirped black solitary waves.

Equations (5)

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i u z - i α u + ( 1 2 - i γ 2 ) 2 u t 2 + ( - sign + i α 2 ) u 2 u + ( n 4 - i α 4 ) u 4 u = 0 ,
u = A tanh ( ν t ) exp { i [ - β z + γ ln sech ( ν t ) ] } ,
u = A tanh ( ν t ) [ 1 + α 3 sech 2 ( ν t ) ] 1 / 2 × exp ( i { - β z + γ ln sech ( ν t ) [ 1 + α 3 sech 2 ( ν t ) ] 1 / 2 } ) ,
d u d z = α 4 A + 2 A - 2 ( 1 - u 2 / A + 2 ) ( 1 - u 2 / A - 2 ) ,
d ϕ d z = α 2 A 2 ( 1 - 1.2 α 4 A 2 / α 2 ) sin ( 2 ϕ ) / 3 ,

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