Abstract

A novel direct perturbation approach to the self-induced Raman effect on dark optical solitons is discussed. The seeming breakdown of the adiabatic assumption found previously is overcome. A complete adiabatic solution is given, confirming previous numerical and analytical results for the Raman blueshift on a more rigorous basis.

© 1998 Optical Society of America

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References

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  1. Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763–915 (1989).
    [CrossRef]
  2. J. P. Keener and D. W. McLaughlin, “Solitons under perturbation,” Phys. Rev. A 16, 777–790 (1977).
    [CrossRef]
  3. J. P. Gordon, “Theory of the soliton self-frequency shift,” Opt. Lett. 11, 662–664 (1986).
    [CrossRef] [PubMed]
  4. F. M. Mitschke and L. F. Mollenauer, “Discovery of the soliton self-frequency shift,” Opt. Lett. 11, 659–661 (1986).
    [CrossRef] [PubMed]
  5. A. M. Weiner, R. N. Thurston, W. J. Tomlinson, J. P. Heritage, D. E. Leaird, E. M. Kirschner, and R. J. Hawkins, “Temporal and spectral self-shifts of dark optical solitons,” Opt. Lett. 14, 868–870 (1989).
    [CrossRef] [PubMed]
  6. W. Zhao and E. Bourkoff, “Generation, propagation, and amplification of dark solitons,” J. Opt. Soc. Am. B 9, 1134–1144 (1992).
    [CrossRef]
  7. Y. S. Kivshar, “Perturbation-induced dynamics of small-amplitude dark optical solitons,” Opt. Lett. 15, 1273–1275 (1990).
    [CrossRef] [PubMed]
  8. I. M. Uzunov and V. S. Gerdjikov, “Self-frequency shift of dark solitons in optical fibers,” Phys. Rev. A 47, 1582–1585 (1993).
    [CrossRef] [PubMed]
  9. Y. S. Kivshar and X. Yang, “Perturbation-induced dynamics of dark solitons,” Phys. Rev. E 49, 1657–1670 (1994).
    [CrossRef]
  10. H. Ikeda, M. M. Matsumoto, and A. Hasegawa, “Transmission control of dark solitons by means of nonlinear gain,” Opt. Lett. 20, 1113–1115 (1995).
    [CrossRef] [PubMed]
  11. S. Burtsev and R. Camassa, “Nonadiabatic dynamics of dark solitons,” J. Opt. Soc. Am. B 14, 1782–1787 (1997).
    [CrossRef]
  12. Y. S. Kivshar and B. L. Davies, “Optical dark solitons: physics and applications,” Phys. Rep. 298, 81–197 (1997).
    [CrossRef]
  13. V. V. Konotop and V. E. Vekslerchik, “Direct perturbation theory for dark solitons,” Phys. Rev. E 49, 2397–2407 (1994).
    [CrossRef]
  14. X. J. Chen, Z. D. Chen, and N. N. Huang, “A direct perturbation theory for dark solitons based on a complete set of the squared Jost solutions,” J. Phys. A 31, 6929–6947 (1998).
    [CrossRef]
  15. L. D. Faddeev and L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons (Springer-Verlag, Berlin, 1987).
  16. D. E. Pelinovsky, Y. S. Kivshar, and V. V. Afanasjev, “Instability-induced dynamics of dark solitons,” Phys. Rev. E 54, 2015–2032 (1996).
    [CrossRef]
  17. D. J. Kaup and A. C. Newell, “Solitons as particles, oscillators, and in slowly changing media: a singular perturbation theory,” Proc. R. Soc. London, Ser. A 361, 413–446 (1978).
    [CrossRef]
  18. V. I. Karpman, “Soliton evolution in the presence of perturbation,” Phys. Scr. 20, 462–478 (1979).
    [CrossRef]

1998 (1)

X. J. Chen, Z. D. Chen, and N. N. Huang, “A direct perturbation theory for dark solitons based on a complete set of the squared Jost solutions,” J. Phys. A 31, 6929–6947 (1998).
[CrossRef]

1997 (2)

S. Burtsev and R. Camassa, “Nonadiabatic dynamics of dark solitons,” J. Opt. Soc. Am. B 14, 1782–1787 (1997).
[CrossRef]

Y. S. Kivshar and B. L. Davies, “Optical dark solitons: physics and applications,” Phys. Rep. 298, 81–197 (1997).
[CrossRef]

1996 (1)

D. E. Pelinovsky, Y. S. Kivshar, and V. V. Afanasjev, “Instability-induced dynamics of dark solitons,” Phys. Rev. E 54, 2015–2032 (1996).
[CrossRef]

1995 (1)

1994 (2)

Y. S. Kivshar and X. Yang, “Perturbation-induced dynamics of dark solitons,” Phys. Rev. E 49, 1657–1670 (1994).
[CrossRef]

V. V. Konotop and V. E. Vekslerchik, “Direct perturbation theory for dark solitons,” Phys. Rev. E 49, 2397–2407 (1994).
[CrossRef]

1993 (1)

I. M. Uzunov and V. S. Gerdjikov, “Self-frequency shift of dark solitons in optical fibers,” Phys. Rev. A 47, 1582–1585 (1993).
[CrossRef] [PubMed]

1992 (1)

1990 (1)

1989 (2)

1986 (2)

1979 (1)

V. I. Karpman, “Soliton evolution in the presence of perturbation,” Phys. Scr. 20, 462–478 (1979).
[CrossRef]

1978 (1)

D. J. Kaup and A. C. Newell, “Solitons as particles, oscillators, and in slowly changing media: a singular perturbation theory,” Proc. R. Soc. London, Ser. A 361, 413–446 (1978).
[CrossRef]

1977 (1)

J. P. Keener and D. W. McLaughlin, “Solitons under perturbation,” Phys. Rev. A 16, 777–790 (1977).
[CrossRef]

Afanasjev, V. V.

D. E. Pelinovsky, Y. S. Kivshar, and V. V. Afanasjev, “Instability-induced dynamics of dark solitons,” Phys. Rev. E 54, 2015–2032 (1996).
[CrossRef]

Bourkoff, E.

Burtsev, S.

Camassa, R.

Chen, X. J.

X. J. Chen, Z. D. Chen, and N. N. Huang, “A direct perturbation theory for dark solitons based on a complete set of the squared Jost solutions,” J. Phys. A 31, 6929–6947 (1998).
[CrossRef]

Chen, Z. D.

X. J. Chen, Z. D. Chen, and N. N. Huang, “A direct perturbation theory for dark solitons based on a complete set of the squared Jost solutions,” J. Phys. A 31, 6929–6947 (1998).
[CrossRef]

Davies, B. L.

Y. S. Kivshar and B. L. Davies, “Optical dark solitons: physics and applications,” Phys. Rep. 298, 81–197 (1997).
[CrossRef]

Gerdjikov, V. S.

I. M. Uzunov and V. S. Gerdjikov, “Self-frequency shift of dark solitons in optical fibers,” Phys. Rev. A 47, 1582–1585 (1993).
[CrossRef] [PubMed]

Gordon, J. P.

Hasegawa, A.

Hawkins, R. J.

Heritage, J. P.

Huang, N. N.

X. J. Chen, Z. D. Chen, and N. N. Huang, “A direct perturbation theory for dark solitons based on a complete set of the squared Jost solutions,” J. Phys. A 31, 6929–6947 (1998).
[CrossRef]

Ikeda, H.

Karpman, V. I.

V. I. Karpman, “Soliton evolution in the presence of perturbation,” Phys. Scr. 20, 462–478 (1979).
[CrossRef]

Kaup, D. J.

D. J. Kaup and A. C. Newell, “Solitons as particles, oscillators, and in slowly changing media: a singular perturbation theory,” Proc. R. Soc. London, Ser. A 361, 413–446 (1978).
[CrossRef]

Keener, J. P.

J. P. Keener and D. W. McLaughlin, “Solitons under perturbation,” Phys. Rev. A 16, 777–790 (1977).
[CrossRef]

Kirschner, E. M.

Kivshar, Y. S.

Y. S. Kivshar and B. L. Davies, “Optical dark solitons: physics and applications,” Phys. Rep. 298, 81–197 (1997).
[CrossRef]

D. E. Pelinovsky, Y. S. Kivshar, and V. V. Afanasjev, “Instability-induced dynamics of dark solitons,” Phys. Rev. E 54, 2015–2032 (1996).
[CrossRef]

Y. S. Kivshar and X. Yang, “Perturbation-induced dynamics of dark solitons,” Phys. Rev. E 49, 1657–1670 (1994).
[CrossRef]

Y. S. Kivshar, “Perturbation-induced dynamics of small-amplitude dark optical solitons,” Opt. Lett. 15, 1273–1275 (1990).
[CrossRef] [PubMed]

Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763–915 (1989).
[CrossRef]

Konotop, V. V.

V. V. Konotop and V. E. Vekslerchik, “Direct perturbation theory for dark solitons,” Phys. Rev. E 49, 2397–2407 (1994).
[CrossRef]

Leaird, D. E.

Malomed, B. A.

Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763–915 (1989).
[CrossRef]

Matsumoto, M. M.

McLaughlin, D. W.

J. P. Keener and D. W. McLaughlin, “Solitons under perturbation,” Phys. Rev. A 16, 777–790 (1977).
[CrossRef]

Mitschke, F. M.

Mollenauer, L. F.

Newell, A. C.

D. J. Kaup and A. C. Newell, “Solitons as particles, oscillators, and in slowly changing media: a singular perturbation theory,” Proc. R. Soc. London, Ser. A 361, 413–446 (1978).
[CrossRef]

Pelinovsky, D. E.

D. E. Pelinovsky, Y. S. Kivshar, and V. V. Afanasjev, “Instability-induced dynamics of dark solitons,” Phys. Rev. E 54, 2015–2032 (1996).
[CrossRef]

Thurston, R. N.

Tomlinson, W. J.

Uzunov, I. M.

I. M. Uzunov and V. S. Gerdjikov, “Self-frequency shift of dark solitons in optical fibers,” Phys. Rev. A 47, 1582–1585 (1993).
[CrossRef] [PubMed]

Vekslerchik, V. E.

V. V. Konotop and V. E. Vekslerchik, “Direct perturbation theory for dark solitons,” Phys. Rev. E 49, 2397–2407 (1994).
[CrossRef]

Weiner, A. M.

Yang, X.

Y. S. Kivshar and X. Yang, “Perturbation-induced dynamics of dark solitons,” Phys. Rev. E 49, 1657–1670 (1994).
[CrossRef]

Zhao, W.

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

J. Phys. A (1)

X. J. Chen, Z. D. Chen, and N. N. Huang, “A direct perturbation theory for dark solitons based on a complete set of the squared Jost solutions,” J. Phys. A 31, 6929–6947 (1998).
[CrossRef]

Opt. Lett. (5)

Phys. Rep. (1)

Y. S. Kivshar and B. L. Davies, “Optical dark solitons: physics and applications,” Phys. Rep. 298, 81–197 (1997).
[CrossRef]

Phys. Rev. A (2)

J. P. Keener and D. W. McLaughlin, “Solitons under perturbation,” Phys. Rev. A 16, 777–790 (1977).
[CrossRef]

I. M. Uzunov and V. S. Gerdjikov, “Self-frequency shift of dark solitons in optical fibers,” Phys. Rev. A 47, 1582–1585 (1993).
[CrossRef] [PubMed]

Phys. Rev. E (3)

Y. S. Kivshar and X. Yang, “Perturbation-induced dynamics of dark solitons,” Phys. Rev. E 49, 1657–1670 (1994).
[CrossRef]

V. V. Konotop and V. E. Vekslerchik, “Direct perturbation theory for dark solitons,” Phys. Rev. E 49, 2397–2407 (1994).
[CrossRef]

D. E. Pelinovsky, Y. S. Kivshar, and V. V. Afanasjev, “Instability-induced dynamics of dark solitons,” Phys. Rev. E 54, 2015–2032 (1996).
[CrossRef]

Phys. Scr. (1)

V. I. Karpman, “Soliton evolution in the presence of perturbation,” Phys. Scr. 20, 462–478 (1979).
[CrossRef]

Proc. R. Soc. London, Ser. A (1)

D. J. Kaup and A. C. Newell, “Solitons as particles, oscillators, and in slowly changing media: a singular perturbation theory,” Proc. R. Soc. London, Ser. A 361, 413–446 (1978).
[CrossRef]

Rev. Mod. Phys. (1)

Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763–915 (1989).
[CrossRef]

Other (1)

L. D. Faddeev and L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons (Springer-Verlag, Berlin, 1987).

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Equations (47)

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iAx-σ(1/2)Att+|A|2A=0,
r[A]=A(|A|2)t
iux-(1/2)utt+(|u|2-ρ2)u=0.
u1(t, x)=exp(-iβ1)(λ1+ik1 tanh θ1),
ivx-(1/2)vtt+(|v|2-ρ2)v=r[v].
iqx-½qTT+(2|u|2-ρ2)q+u2q¯=R[u],
-iq¯x-½q¯TT+(2|u|2-ρ2)q¯+u¯2q=R[u]¯.
[ix-L(T)]q=R,
L(T)=L11L12L21L22
L11=λ1T+½TT-(2|u|2-ρ2),
L12=-u2,L21=u¯2,
L22=λ1T-½TT+(2|u|2-ρ2),
q=qq¯,R=R-R¯.
ϕ1(T, ζ)=ζ-ζ1ζ-ζ¯1 exp(-iβ1-iκT)+i 1ζ-ζ¯1 k1×sech θ1 exp(-iβ1-θ1-iκT),
ϕ2(T, ζ)=iρζ-1 ζ-ζ1ζ-ζ¯1 exp(-iβ1-iκT)-1ζ-ζ¯1 k1 sech θ1 exp(-θ1-iκT);
ψ1(T, ζ)=-iρζ-1 exp(iκT)-1ζ-ζ¯1 k1 sech θ1×exp(-iβ1-θ1+iκT),
ψ2(T, ζ)=exp(iκT)-i 1ζ-ζ¯1 k1 sech θ1×exp(-θ1+iκT).
Φ(T, ζ)=ϕ12(T, ζ)ϕ22(T, ζ),Ψ(T, ζ)=ψ12(T, ζ)ψ22(T, ζ),
L(T)Φ(T, ζ)=2κ(λ-λ1)Φ(T, ζ),
L(T)Φ(T, ζ1)=0,
L(T)Φ˙(T, ζ1)=2k12ζ1-1Φ(T, ζ1),
Φ(T, ζ)|Φ(T, ζ)=-+dTΦ(T, ζ)AΦ(T, ζ),
Φ(ζ)|Φ(ζ)=-2πa2(ζ)(1-ρ2ζ-2)δ(ζ-ζ),
Φ(ζ1)|Φ(ζ1)=0,
Φ(ζ1)|Φ˙(ζ1)=Φ˙(ζ1)|Φ(ζ1)=2k1ζ1-1a˙2(ζ1),
Φ˙(ζ1)|Φ˙(ζ1)=2a˙(ζ1)a¨(ζ1)k1ζ1-1-i2a˙2(ζ1)ρ2ζ1-3.
a(ζ)=exp(-iβ1) ζ-ζ1ζ-ζ¯1.
|q=-12π -+dζq(ζ)|Φ(ζ)+q1|Φ(ζ1)+q2|Φ˙(ζ1),
iq2xΦ(ζ1)|Φ˙(ζ1)=Φ(ζ1)|R,
(iq1x-2k1ζ1-1q2)Φ˙(ζ1)|Φ(ζ1)+iq2xΦ˙(ζ1)|Φ(ζ1)
=Φ˙(ζ1)|R,
iqx(ζ)+2κ(λ-λ1)q(ζ)=ζ2a2(ζ) κ-1Φ(ζ)|R.
q2=-i Φ(ζ1)|RΦ(ζ1)|Φ˙(ζ1) x
Φ(ζ1)|R=0,
q1=-i Φ˙(ζ1)|RΦ˙(ζ1)|Φ(ζ1) x,
Φ˙(ζ1)|R=0
q(T, x)=-12π -+dζq(ζ, x)ϕ12(T, ζ),
λ=12 -+dθ1 sech2 θ1 Im[exp(iβ1)r],
-2Lk1ρ-2 k13ρ Tc
=λ1ρ -+dθ1θ1 sech2 θ1 Im[exp(iβ1)r]--+dθ1 sech θ1 exp(-θ1)Im[exp(i2β1)r],
r[u]=u1(|u1|2)T=2k13 sech2 θ1 tanh θ1u1,
Im[exp(iβ1)r]=2k14 sech2 θ1 tanh2 θ1,
Im[exp(i2β1)r]=2 k14λ1ρ sech3 θ1 tanh θ1 exp(θ1).
λ1(x)=λ1(0)+(4/15)k14(0)x,
k1(x)=k1(0)-(4/15)λ1(0)k13(0)x,
β1(x)=β1(0)-(4/15)k13(0)x,
ρ(x)=ρ(0),Tc=0.

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