Abstract

Soliton propagation along optical fibers is discussed on the basis of the modified nonlinear Schrödinger equation, which is known to be completely integrable in terms of the inverse-scattering transformation. Numerical observation of soliton propagation reveals various kinds of dynamical behavior, which cannot be accounted for by treating the nonlinear derivative term of the modified nonlinear Schrödinger equation as a perturbation term for the cubic nonlinear Schrödinger equation.

© 1987 Optical Society of America

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References

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  1. A. Hasegawa, F. Tappert, Appl. Phys. Lett. 23, 142 (1973).
    [CrossRef]
  2. L. F. Mollenauer, R. H. Stolen, J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980).
    [CrossRef]
  3. H. Nakatsuka, D. Grischkowsky, A. C. Balant, Phys. Rev. Lett. 47, 910 (1981).
    [CrossRef]
  4. N. Tzoar, M. Jain, Phys. Rev. A 23, 1266 (1981).
    [CrossRef]
  5. Y. Kodama, A. Hasegawa, IEEE J. Quantum Electron.QE-23 (to be published).
  6. E. A. Golovchenko, E. M. Dianov, A. M. Prokhorov, V. N. Serkin, Pis'ma Zh. Eksp. Teor. Fiz. 42, 74 (1985)[JETP Lett. 42, 88 (1985)].
  7. D. Anderson, M. Lisak, Phys. Rev. A 27, 1393 (1983).
    [CrossRef]
  8. F. M. Mitschke, L. F. Mollenauer, Opt. Lett. 11, 659 (1986).
    [CrossRef] [PubMed]
  9. J. P. Gordon, Opt. Lett. 11, 662 (1986).
    [CrossRef] [PubMed]
  10. E. Mjolhus, J. Plasma Phys. 16, 321 (1976).
    [CrossRef]
  11. K. Mio, T. Ogino, K. Minami, S. Takeda, J. Phys. Soc. Jpn. 41, 265 (1976).
    [CrossRef]
  12. M. Wadati, K. Konno, Y. H. Ichikawa, J. Phys. Soc. Jpn. 46, 1965 (1979).
    [CrossRef]
  13. Y. H. Ichikawa, K. Konno, M. Wadati, H. Sanuki, J. Phys. Soc. Jpn. 48, 279 (1980).
    [CrossRef]
  14. J. Satsuma, N. Yajima, Progr. Theor. Phys. Suppl. 55, 284 (1974).
    [CrossRef]
  15. J. Weiland, Y. H. Ichikawa, H. Wilhelmsson, Phys. Scr. 17, 518 (1978).
    [CrossRef]

1986 (2)

1985 (1)

E. A. Golovchenko, E. M. Dianov, A. M. Prokhorov, V. N. Serkin, Pis'ma Zh. Eksp. Teor. Fiz. 42, 74 (1985)[JETP Lett. 42, 88 (1985)].

1983 (1)

D. Anderson, M. Lisak, Phys. Rev. A 27, 1393 (1983).
[CrossRef]

1981 (2)

H. Nakatsuka, D. Grischkowsky, A. C. Balant, Phys. Rev. Lett. 47, 910 (1981).
[CrossRef]

N. Tzoar, M. Jain, Phys. Rev. A 23, 1266 (1981).
[CrossRef]

1980 (2)

L. F. Mollenauer, R. H. Stolen, J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980).
[CrossRef]

Y. H. Ichikawa, K. Konno, M. Wadati, H. Sanuki, J. Phys. Soc. Jpn. 48, 279 (1980).
[CrossRef]

1979 (1)

M. Wadati, K. Konno, Y. H. Ichikawa, J. Phys. Soc. Jpn. 46, 1965 (1979).
[CrossRef]

1978 (1)

J. Weiland, Y. H. Ichikawa, H. Wilhelmsson, Phys. Scr. 17, 518 (1978).
[CrossRef]

1976 (2)

E. Mjolhus, J. Plasma Phys. 16, 321 (1976).
[CrossRef]

K. Mio, T. Ogino, K. Minami, S. Takeda, J. Phys. Soc. Jpn. 41, 265 (1976).
[CrossRef]

1974 (1)

J. Satsuma, N. Yajima, Progr. Theor. Phys. Suppl. 55, 284 (1974).
[CrossRef]

1973 (1)

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

Anderson, D.

D. Anderson, M. Lisak, Phys. Rev. A 27, 1393 (1983).
[CrossRef]

Balant, A. C.

H. Nakatsuka, D. Grischkowsky, A. C. Balant, Phys. Rev. Lett. 47, 910 (1981).
[CrossRef]

Dianov, E. M.

E. A. Golovchenko, E. M. Dianov, A. M. Prokhorov, V. N. Serkin, Pis'ma Zh. Eksp. Teor. Fiz. 42, 74 (1985)[JETP Lett. 42, 88 (1985)].

Golovchenko, E. A.

E. A. Golovchenko, E. M. Dianov, A. M. Prokhorov, V. N. Serkin, Pis'ma Zh. Eksp. Teor. Fiz. 42, 74 (1985)[JETP Lett. 42, 88 (1985)].

Gordon, J. P.

J. P. Gordon, Opt. Lett. 11, 662 (1986).
[CrossRef] [PubMed]

L. F. Mollenauer, R. H. Stolen, J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980).
[CrossRef]

Grischkowsky, D.

H. Nakatsuka, D. Grischkowsky, A. C. Balant, Phys. Rev. Lett. 47, 910 (1981).
[CrossRef]

Hasegawa, A.

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

Y. Kodama, A. Hasegawa, IEEE J. Quantum Electron.QE-23 (to be published).

Ichikawa, Y. H.

Y. H. Ichikawa, K. Konno, M. Wadati, H. Sanuki, J. Phys. Soc. Jpn. 48, 279 (1980).
[CrossRef]

M. Wadati, K. Konno, Y. H. Ichikawa, J. Phys. Soc. Jpn. 46, 1965 (1979).
[CrossRef]

J. Weiland, Y. H. Ichikawa, H. Wilhelmsson, Phys. Scr. 17, 518 (1978).
[CrossRef]

Jain, M.

N. Tzoar, M. Jain, Phys. Rev. A 23, 1266 (1981).
[CrossRef]

Kodama, Y.

Y. Kodama, A. Hasegawa, IEEE J. Quantum Electron.QE-23 (to be published).

Konno, K.

Y. H. Ichikawa, K. Konno, M. Wadati, H. Sanuki, J. Phys. Soc. Jpn. 48, 279 (1980).
[CrossRef]

M. Wadati, K. Konno, Y. H. Ichikawa, J. Phys. Soc. Jpn. 46, 1965 (1979).
[CrossRef]

Lisak, M.

D. Anderson, M. Lisak, Phys. Rev. A 27, 1393 (1983).
[CrossRef]

Minami, K.

K. Mio, T. Ogino, K. Minami, S. Takeda, J. Phys. Soc. Jpn. 41, 265 (1976).
[CrossRef]

Mio, K.

K. Mio, T. Ogino, K. Minami, S. Takeda, J. Phys. Soc. Jpn. 41, 265 (1976).
[CrossRef]

Mitschke, F. M.

Mjolhus, E.

E. Mjolhus, J. Plasma Phys. 16, 321 (1976).
[CrossRef]

Mollenauer, L. F.

F. M. Mitschke, L. F. Mollenauer, Opt. Lett. 11, 659 (1986).
[CrossRef] [PubMed]

L. F. Mollenauer, R. H. Stolen, J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980).
[CrossRef]

Nakatsuka, H.

H. Nakatsuka, D. Grischkowsky, A. C. Balant, Phys. Rev. Lett. 47, 910 (1981).
[CrossRef]

Ogino, T.

K. Mio, T. Ogino, K. Minami, S. Takeda, J. Phys. Soc. Jpn. 41, 265 (1976).
[CrossRef]

Prokhorov, A. M.

E. A. Golovchenko, E. M. Dianov, A. M. Prokhorov, V. N. Serkin, Pis'ma Zh. Eksp. Teor. Fiz. 42, 74 (1985)[JETP Lett. 42, 88 (1985)].

Sanuki, H.

Y. H. Ichikawa, K. Konno, M. Wadati, H. Sanuki, J. Phys. Soc. Jpn. 48, 279 (1980).
[CrossRef]

Satsuma, J.

J. Satsuma, N. Yajima, Progr. Theor. Phys. Suppl. 55, 284 (1974).
[CrossRef]

Serkin, V. N.

E. A. Golovchenko, E. M. Dianov, A. M. Prokhorov, V. N. Serkin, Pis'ma Zh. Eksp. Teor. Fiz. 42, 74 (1985)[JETP Lett. 42, 88 (1985)].

Stolen, R. H.

L. F. Mollenauer, R. H. Stolen, J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980).
[CrossRef]

Takeda, S.

K. Mio, T. Ogino, K. Minami, S. Takeda, J. Phys. Soc. Jpn. 41, 265 (1976).
[CrossRef]

Tappert, F.

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

Tzoar, N.

N. Tzoar, M. Jain, Phys. Rev. A 23, 1266 (1981).
[CrossRef]

Wadati, M.

Y. H. Ichikawa, K. Konno, M. Wadati, H. Sanuki, J. Phys. Soc. Jpn. 48, 279 (1980).
[CrossRef]

M. Wadati, K. Konno, Y. H. Ichikawa, J. Phys. Soc. Jpn. 46, 1965 (1979).
[CrossRef]

Weiland, J.

J. Weiland, Y. H. Ichikawa, H. Wilhelmsson, Phys. Scr. 17, 518 (1978).
[CrossRef]

Wilhelmsson, H.

J. Weiland, Y. H. Ichikawa, H. Wilhelmsson, Phys. Scr. 17, 518 (1978).
[CrossRef]

Yajima, N.

J. Satsuma, N. Yajima, Progr. Theor. Phys. Suppl. 55, 284 (1974).
[CrossRef]

Appl. Phys. Lett. (1)

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

J. Phys. Soc. Jpn. (3)

K. Mio, T. Ogino, K. Minami, S. Takeda, J. Phys. Soc. Jpn. 41, 265 (1976).
[CrossRef]

M. Wadati, K. Konno, Y. H. Ichikawa, J. Phys. Soc. Jpn. 46, 1965 (1979).
[CrossRef]

Y. H. Ichikawa, K. Konno, M. Wadati, H. Sanuki, J. Phys. Soc. Jpn. 48, 279 (1980).
[CrossRef]

J. Plasma Phys. (1)

E. Mjolhus, J. Plasma Phys. 16, 321 (1976).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. A (2)

D. Anderson, M. Lisak, Phys. Rev. A 27, 1393 (1983).
[CrossRef]

N. Tzoar, M. Jain, Phys. Rev. A 23, 1266 (1981).
[CrossRef]

Phys. Rev. Lett. (2)

L. F. Mollenauer, R. H. Stolen, J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980).
[CrossRef]

H. Nakatsuka, D. Grischkowsky, A. C. Balant, Phys. Rev. Lett. 47, 910 (1981).
[CrossRef]

Phys. Scr. (1)

J. Weiland, Y. H. Ichikawa, H. Wilhelmsson, Phys. Scr. 17, 518 (1978).
[CrossRef]

Pis'ma Zh. Eksp. Teor. Fiz. (1)

E. A. Golovchenko, E. M. Dianov, A. M. Prokhorov, V. N. Serkin, Pis'ma Zh. Eksp. Teor. Fiz. 42, 74 (1985)[JETP Lett. 42, 88 (1985)].

Progr. Theor. Phys. Suppl. (1)

J. Satsuma, N. Yajima, Progr. Theor. Phys. Suppl. 55, 284 (1974).
[CrossRef]

Other (1)

Y. Kodama, A. Hasegawa, IEEE J. Quantum Electron.QE-23 (to be published).

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Figures (7)

Fig. 1
Fig. 1

Spatial evolution of the initial pulse q(τ, ξ = 0) = sech(τ) for the parameters R = 1 and (I) γ = 0.25, (II) γ = 0.5, and (III) γ = 1 at (a) ξ = 0, (b) ξ = π/2, (c) ξ = π, (d) ξ = 3/2π, and (e) ξ = 5π. The lines inside the pulses represent the real part of q.

Fig. 2
Fig. 2

Time delay τ0(ξ) of the pulse peak at different positions of ξ for R = 1 and (a) γ = 0.1, (b) γ = 0.25, and (c) γ = 0.5.

Fig. 3
Fig. 3

Spatial evolution of the peak height ρ0(τ0) for R = 1 and γ = 0.5.

Fig. 4
Fig. 4

Spatial evolution of initial pulse q(τ, ξ = 0) = sech(τ) for R = 4 and γ = 0.1, which stands for the two-soliton bound state of the cubic nonlinear Schrödinger equation.

Fig. 5
Fig. 5

Spatial evolution o the initial pulse q(τ, ξ = 0) = sech(τ) for R = 9 and γ = 0.25, which stands for the three-soliton bound state of the cubic nonlinear Schrödinger equation.

Fig. 6
Fig. 6

Spatial growth of the three peaks observed at the interval of Δξ = π for R = 9 and γ = 0.1. The heights of the peaks approach the asymptotic values of ρ1 = 0.333, ρ2 = 1.000, and ρ3 = 1.666.

Fig. 7
Fig. 7

Time delay τ(ξ) of the three peaks observed at the interval of Δξ = π for R = 9 and γ = 0.1. The rate of delay M is M1 = 0.0331, M2 = 0.0999, and M3 = 0.1667.

Equations (9)

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i ξ q 1 2 2 τ 2 q R | q | 2 q + i γ τ ( | q | 2 q ) = 0
q ( τ , ξ ) = ρ ( z ) exp { i [ K ξ Ω τ θ ( z ) ] } ,
Ω = M ,
K = ½ M 2 ½ ( R γ M ) ρ 0 2 γ 2 ρ 0 4 .
ρ ( z ) = ρ 0 2 ν [ cosh 2 ( μ z ) + ν 1 2 ν ] 1 / 2
θ ( z ) = 3 2 μ γ ρ 0 2 1 ν tan 1 [ 1 ν tanh ( μ z ) ] ,
μ 2 = ( R γ M ) ρ 0 2 + γ 2 ρ 0 4 ,
ν = ( R γ M ) ρ 0 2 / μ 2 .
q ( τ , ξ ) = 2 λ I R sech [ 2 λ I ( τ 2 λ R ξ ) ] × exp { 2 i [ λ R τ + ( λ R 2 λ I 2 ) ξ ] } ,

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