Abstract

The propagation of solitons in materials with saturable nonlinearity has been investigated with numerical and analytical methods. It is demonstrated that bistable (or two-state) solitons exist that describe pulses with the same duration but different peak powers. It is proved that both solution branches are stable. The minimum possible pulse duration of a fundamental soliton depending on the saturation parameter has been predicted. The influence of loss and the evolution of high-order solitons are studied.

© 1991 Optical Society of America

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  1. A. Hasegawa and F. D. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142 (1973).
    [Crossref]
  2. 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 (1980).
    [Crossref]
  3. L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Extreme picosecond pulse narrowing by means of soliton effect in single-mode optical fibers,” Opt. Lett. 8, 289 (1983).
    [Crossref] [PubMed]
  4. A. Hasegawa, Optical Solitons in Fibers, Vol. 116 of Springer Tracts in Modern Physics (Springer-Verlag, Berlin, 1989).
    [Crossref]
  5. N. Tzoar and M. Jain, “Self-phase modulation in long-geometry optical waveguides,” Phys. Rev. A 23, 1266 (1981).
    [Crossref]
  6. D. Yevick and B. Hermanson, “Soliton analysis with the propagating beam method,” Opt. Commun. 47, 101 (1983).
    [Crossref]
  7. A. D. Boardman and G. S. Cooper, Appl. Sci. Res. 41, 333 (1984).
    [Crossref]
  8. D. N. Christodoulides and R. J. Joseph, “Femtosecond solitary waves in optical fibers—beyond the slowly varying envelope approximation,” Appl. Phys. Lett. 47, 76 (1985).
    [Crossref]
  9. E. A. Golovchenko, E. M. Dianov, As. M. Prokhorov, and V. N. Serkin, “Decay of optical solitons,” Pis’ma Zh. Eksp. Teor. Fiz. 42, 74 (1985) [JETP Lett. 42, 87 (1985)].
  10. J. P. Gordon, “Theory of the soliton self-frequency shift,” Opt. Lett. 11, 662 (1986).
    [Crossref] [PubMed]
  11. K. Okkuma, K. H. Ichikawa, and K. Abe, “Soliton propagation along optical fibers,” Opt. Lett. 12, 516 (1987).
    [Crossref]
  12. W. Hodel and W. P. Weber, “Decay of femtosecond higher-order solitons in an optical fiber induced by Raman self-pumping,” Opt. Lett. 12, 924 (1987).
    [Crossref] [PubMed]
  13. P. K. A. Way, C. R. Menyk, Y. C. Lee, and H. Chen, “Nonlinear pulse propagation in the neighborhood of the zero dispersion wavelength of monomode optical fibers,” Opt. Lett. 11, 464 (1988).
    [Crossref]
  14. G. Agrawal, Nonlinear Fiber Optics (Academic, Boston, Mass., 1983).
  15. A. A. Kolokolov, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 17, 132 (1974) [Radiophys. Quantum Electron. 17, 1016 (1976)].
  16. E. W. Laedke, K. H. Spatschek, and L. J. Stenflo, “Evolution theorem for a class of perturbed envelope soliton solutions,” J. Math. Phys. 24, 2764 (1987).
    [Crossref]
  17. P. Roussignol, D. Richard, J. Lukasik, and C. Flytzanis, “New results on optical phase conjugation in semiconductor-doped glasses,” J. Opt. Soc. Am. B 4, 5 (1987).
    [Crossref]
  18. J.-L. Coutaz and M. Kull, “Saturation of the nonlinear index of refraction in semiconductor glasses,” J. Opt. Soc. Am. B 8, 99 (1991).
    [Crossref]
  19. G. S. Stegman and R. H. Stolen “Waveguides and fibers for nonlinear optics,” J. Opt. Soc. Am. B 6, 652 (1989).
    [Crossref]
  20. A. E. Kaplan, “Bistable solitons,” Phys. Rev. Lett. 55, 1291 (1985).
    [Crossref] [PubMed]
  21. R. A. Enns and S. S. Rangnekar, “Bistable solitons and optical switching,” IEEE J. Quantum Electron. QE-23, 1199 (1987).
    [Crossref]
  22. R. A. Enns, S. S. Rangnekar, and A. E. Kaplan, “Robust bistable solitons of the highly nonlinear Schrödinger equation,” Phys. Rev. A 35, 466 (1987).
    [Crossref] [PubMed]
  23. J. Herrman and B. Wilhelmi, “Influence of various nonlinear optical processes on femtosecond light pulse propagation in fibers, E. Klose and B. Wilhelmi, eds., in Ultrafast Phenomena in Spectroscopy, Vol. 49 of Springer Proceedings in Physics (Springer-Verlag, Berlin, 1990), p. 157.
    [Crossref]
  24. D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 6, 3135 (1983).
    [Crossref]
  25. J. Herrmann, “Propagation of ultrashort light pulses in fibers with saturable nonlinearity in the normal-dispersion regime,” J. Opt. Soc. Am. B 8, 1507 (1991).
    [Crossref]
  26. A. S. Gouveia-Neto, A. S. Gomes, and J. R. Taylor, “Generation of 33 fsec pulse at 1.32 μ m through a high-order soliton effect in a single-mode optical fiber,” Opt. Lett. 12, 395 (1987).
    [Crossref] [PubMed]
  27. A. S. Gouveia-Neto, A. S. Gomes, and J. R. Taylor, “Pulses of four optical cycles from on optimized optical fibre grating pair soliton pulse compressor at 1.32 μ m,” J. Mod. Opt. 35, 7 (1988).
    [Crossref]

1991 (2)

1989 (1)

1988 (2)

A. S. Gouveia-Neto, A. S. Gomes, and J. R. Taylor, “Pulses of four optical cycles from on optimized optical fibre grating pair soliton pulse compressor at 1.32 μ m,” J. Mod. Opt. 35, 7 (1988).
[Crossref]

P. K. A. Way, C. R. Menyk, Y. C. Lee, and H. Chen, “Nonlinear pulse propagation in the neighborhood of the zero dispersion wavelength of monomode optical fibers,” Opt. Lett. 11, 464 (1988).
[Crossref]

1987 (7)

1986 (1)

1985 (3)

D. N. Christodoulides and R. J. Joseph, “Femtosecond solitary waves in optical fibers—beyond the slowly varying envelope approximation,” Appl. Phys. Lett. 47, 76 (1985).
[Crossref]

E. A. Golovchenko, E. M. Dianov, As. M. Prokhorov, and V. N. Serkin, “Decay of optical solitons,” Pis’ma Zh. Eksp. Teor. Fiz. 42, 74 (1985) [JETP Lett. 42, 87 (1985)].

A. E. Kaplan, “Bistable solitons,” Phys. Rev. Lett. 55, 1291 (1985).
[Crossref] [PubMed]

1984 (1)

A. D. Boardman and G. S. Cooper, Appl. Sci. Res. 41, 333 (1984).
[Crossref]

1983 (3)

D. Yevick and B. Hermanson, “Soliton analysis with the propagating beam method,” Opt. Commun. 47, 101 (1983).
[Crossref]

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

D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 6, 3135 (1983).
[Crossref]

1981 (1)

N. Tzoar and M. Jain, “Self-phase modulation in long-geometry optical waveguides,” Phys. Rev. A 23, 1266 (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 (1980).
[Crossref]

1974 (1)

A. A. Kolokolov, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 17, 132 (1974) [Radiophys. Quantum Electron. 17, 1016 (1976)].

1973 (1)

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

Abe, K.

Agrawal, G.

G. Agrawal, Nonlinear Fiber Optics (Academic, Boston, Mass., 1983).

Anderson, D.

D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 6, 3135 (1983).
[Crossref]

Boardman, A. D.

A. D. Boardman and G. S. Cooper, Appl. Sci. Res. 41, 333 (1984).
[Crossref]

Chen, H.

Christodoulides, D. N.

D. N. Christodoulides and R. J. Joseph, “Femtosecond solitary waves in optical fibers—beyond the slowly varying envelope approximation,” Appl. Phys. Lett. 47, 76 (1985).
[Crossref]

Cooper, G. S.

A. D. Boardman and G. S. Cooper, Appl. Sci. Res. 41, 333 (1984).
[Crossref]

Coutaz, J.-L.

Dianov, E. M.

E. A. Golovchenko, E. M. Dianov, As. M. Prokhorov, and V. N. Serkin, “Decay of optical solitons,” Pis’ma Zh. Eksp. Teor. Fiz. 42, 74 (1985) [JETP Lett. 42, 87 (1985)].

Enns, R. A.

R. A. Enns, S. S. Rangnekar, and A. E. Kaplan, “Robust bistable solitons of the highly nonlinear Schrödinger equation,” Phys. Rev. A 35, 466 (1987).
[Crossref] [PubMed]

R. A. Enns and S. S. Rangnekar, “Bistable solitons and optical switching,” IEEE J. Quantum Electron. QE-23, 1199 (1987).
[Crossref]

Flytzanis, C.

Golovchenko, E. A.

E. A. Golovchenko, E. M. Dianov, As. M. Prokhorov, and V. N. Serkin, “Decay of optical solitons,” Pis’ma Zh. Eksp. Teor. Fiz. 42, 74 (1985) [JETP Lett. 42, 87 (1985)].

Gomes, A. S.

A. S. Gouveia-Neto, A. S. Gomes, and J. R. Taylor, “Pulses of four optical cycles from on optimized optical fibre grating pair soliton pulse compressor at 1.32 μ m,” J. Mod. Opt. 35, 7 (1988).
[Crossref]

A. S. Gouveia-Neto, A. S. Gomes, and J. R. Taylor, “Generation of 33 fsec pulse at 1.32 μ m through a high-order soliton effect in a single-mode optical fiber,” Opt. Lett. 12, 395 (1987).
[Crossref] [PubMed]

Gordon, J. P.

Gouveia-Neto, A. S.

A. S. Gouveia-Neto, A. S. Gomes, and J. R. Taylor, “Pulses of four optical cycles from on optimized optical fibre grating pair soliton pulse compressor at 1.32 μ m,” J. Mod. Opt. 35, 7 (1988).
[Crossref]

A. S. Gouveia-Neto, A. S. Gomes, and J. R. Taylor, “Generation of 33 fsec pulse at 1.32 μ m through a high-order soliton effect in a single-mode optical fiber,” Opt. Lett. 12, 395 (1987).
[Crossref] [PubMed]

Hasegawa, A.

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

A. Hasegawa, Optical Solitons in Fibers, Vol. 116 of Springer Tracts in Modern Physics (Springer-Verlag, Berlin, 1989).
[Crossref]

Hermanson, B.

D. Yevick and B. Hermanson, “Soliton analysis with the propagating beam method,” Opt. Commun. 47, 101 (1983).
[Crossref]

Herrman, J.

J. Herrman and B. Wilhelmi, “Influence of various nonlinear optical processes on femtosecond light pulse propagation in fibers, E. Klose and B. Wilhelmi, eds., in Ultrafast Phenomena in Spectroscopy, Vol. 49 of Springer Proceedings in Physics (Springer-Verlag, Berlin, 1990), p. 157.
[Crossref]

Herrmann, J.

Hodel, W.

Ichikawa, K. H.

Jain, M.

N. Tzoar and M. Jain, “Self-phase modulation in long-geometry optical waveguides,” Phys. Rev. A 23, 1266 (1981).
[Crossref]

Joseph, R. J.

D. N. Christodoulides and R. J. Joseph, “Femtosecond solitary waves in optical fibers—beyond the slowly varying envelope approximation,” Appl. Phys. Lett. 47, 76 (1985).
[Crossref]

Kaplan, A. E.

R. A. Enns, S. S. Rangnekar, and A. E. Kaplan, “Robust bistable solitons of the highly nonlinear Schrödinger equation,” Phys. Rev. A 35, 466 (1987).
[Crossref] [PubMed]

A. E. Kaplan, “Bistable solitons,” Phys. Rev. Lett. 55, 1291 (1985).
[Crossref] [PubMed]

Kolokolov, A. A.

A. A. Kolokolov, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 17, 132 (1974) [Radiophys. Quantum Electron. 17, 1016 (1976)].

Kull, M.

Laedke, E. W.

E. W. Laedke, K. H. Spatschek, and L. J. Stenflo, “Evolution theorem for a class of perturbed envelope soliton solutions,” J. Math. Phys. 24, 2764 (1987).
[Crossref]

Lee, Y. C.

Lukasik, J.

Menyk, C. R.

Mollenauer, L. F.

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Extreme picosecond pulse narrowing by means of soliton effect in single-mode optical fibers,” Opt. Lett. 8, 289 (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 (1980).
[Crossref]

Okkuma, K.

Prokhorov, As. M.

E. A. Golovchenko, E. M. Dianov, As. M. Prokhorov, and V. N. Serkin, “Decay of optical solitons,” Pis’ma Zh. Eksp. Teor. Fiz. 42, 74 (1985) [JETP Lett. 42, 87 (1985)].

Rangnekar, S. S.

R. A. Enns and S. S. Rangnekar, “Bistable solitons and optical switching,” IEEE J. Quantum Electron. QE-23, 1199 (1987).
[Crossref]

R. A. Enns, S. S. Rangnekar, and A. E. Kaplan, “Robust bistable solitons of the highly nonlinear Schrödinger equation,” Phys. Rev. A 35, 466 (1987).
[Crossref] [PubMed]

Richard, D.

Roussignol, P.

Serkin, V. N.

E. A. Golovchenko, E. M. Dianov, As. M. Prokhorov, and V. N. Serkin, “Decay of optical solitons,” Pis’ma Zh. Eksp. Teor. Fiz. 42, 74 (1985) [JETP Lett. 42, 87 (1985)].

Spatschek, K. H.

E. W. Laedke, K. H. Spatschek, and L. J. Stenflo, “Evolution theorem for a class of perturbed envelope soliton solutions,” J. Math. Phys. 24, 2764 (1987).
[Crossref]

Stegman, G. S.

Stenflo, L. J.

E. W. Laedke, K. H. Spatschek, and L. J. Stenflo, “Evolution theorem for a class of perturbed envelope soliton solutions,” J. Math. Phys. 24, 2764 (1987).
[Crossref]

Stolen, R. H.

Tappert, F. D.

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

Taylor, J. R.

A. S. Gouveia-Neto, A. S. Gomes, and J. R. Taylor, “Pulses of four optical cycles from on optimized optical fibre grating pair soliton pulse compressor at 1.32 μ m,” J. Mod. Opt. 35, 7 (1988).
[Crossref]

A. S. Gouveia-Neto, A. S. Gomes, and J. R. Taylor, “Generation of 33 fsec pulse at 1.32 μ m through a high-order soliton effect in a single-mode optical fiber,” Opt. Lett. 12, 395 (1987).
[Crossref] [PubMed]

Tzoar, N.

N. Tzoar and M. Jain, “Self-phase modulation in long-geometry optical waveguides,” Phys. Rev. A 23, 1266 (1981).
[Crossref]

Way, P. K. A.

Weber, W. P.

Wilhelmi, B.

J. Herrman and B. Wilhelmi, “Influence of various nonlinear optical processes on femtosecond light pulse propagation in fibers, E. Klose and B. Wilhelmi, eds., in Ultrafast Phenomena in Spectroscopy, Vol. 49 of Springer Proceedings in Physics (Springer-Verlag, Berlin, 1990), p. 157.
[Crossref]

Yevick, D.

D. Yevick and B. Hermanson, “Soliton analysis with the propagating beam method,” Opt. Commun. 47, 101 (1983).
[Crossref]

Appl. Phys. Lett. (2)

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

D. N. Christodoulides and R. J. Joseph, “Femtosecond solitary waves in optical fibers—beyond the slowly varying envelope approximation,” Appl. Phys. Lett. 47, 76 (1985).
[Crossref]

Appl. Sci. Res. (1)

A. D. Boardman and G. S. Cooper, Appl. Sci. Res. 41, 333 (1984).
[Crossref]

IEEE J. Quantum Electron. (1)

R. A. Enns and S. S. Rangnekar, “Bistable solitons and optical switching,” IEEE J. Quantum Electron. QE-23, 1199 (1987).
[Crossref]

Izv. Vyssh. Uchebn. Zaved. Radiofiz. (1)

A. A. Kolokolov, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 17, 132 (1974) [Radiophys. Quantum Electron. 17, 1016 (1976)].

J. Math. Phys. (1)

E. W. Laedke, K. H. Spatschek, and L. J. Stenflo, “Evolution theorem for a class of perturbed envelope soliton solutions,” J. Math. Phys. 24, 2764 (1987).
[Crossref]

J. Mod. Opt. (1)

A. S. Gouveia-Neto, A. S. Gomes, and J. R. Taylor, “Pulses of four optical cycles from on optimized optical fibre grating pair soliton pulse compressor at 1.32 μ m,” J. Mod. Opt. 35, 7 (1988).
[Crossref]

J. Opt. Soc. Am. B (4)

Opt. Commun. (1)

D. Yevick and B. Hermanson, “Soliton analysis with the propagating beam method,” Opt. Commun. 47, 101 (1983).
[Crossref]

Opt. Lett. (6)

Phys. Rev. A (3)

R. A. Enns, S. S. Rangnekar, and A. E. Kaplan, “Robust bistable solitons of the highly nonlinear Schrödinger equation,” Phys. Rev. A 35, 466 (1987).
[Crossref] [PubMed]

D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 6, 3135 (1983).
[Crossref]

N. Tzoar and M. Jain, “Self-phase modulation in long-geometry optical waveguides,” Phys. Rev. A 23, 1266 (1981).
[Crossref]

Phys. Rev. Lett. (2)

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 (1980).
[Crossref]

A. E. Kaplan, “Bistable solitons,” Phys. Rev. Lett. 55, 1291 (1985).
[Crossref] [PubMed]

Pis’ma Zh. Eksp. Teor. Fiz. (1)

E. A. Golovchenko, E. M. Dianov, As. M. Prokhorov, and V. N. Serkin, “Decay of optical solitons,” Pis’ma Zh. Eksp. Teor. Fiz. 42, 74 (1985) [JETP Lett. 42, 87 (1985)].

Other (3)

A. Hasegawa, Optical Solitons in Fibers, Vol. 116 of Springer Tracts in Modern Physics (Springer-Verlag, Berlin, 1989).
[Crossref]

G. Agrawal, Nonlinear Fiber Optics (Academic, Boston, Mass., 1983).

J. Herrman and B. Wilhelmi, “Influence of various nonlinear optical processes on femtosecond light pulse propagation in fibers, E. Klose and B. Wilhelmi, eds., in Ultrafast Phenomena in Spectroscopy, Vol. 49 of Springer Proceedings in Physics (Springer-Verlag, Berlin, 1990), p. 157.
[Crossref]

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Figures (11)

Fig. 1
Fig. 1

Dependence of the soliton amplitude B on the saturation parameter γ = P0/Isat = |k1″|/(1.76)2τ02κIsat. Curve 1, numerical result; curves 2 and 3, analytical approximations.

Fig. 2
Fig. 2

Soliton shapes for different saturation parameters γ.

Fig. 3
Fig. 3

Normalized pulse energy ˜ relative to the propagation parameter δ = βγ.

Fig. 4
Fig. 4

Normalized intensity of the fundamental soliton relative to the pulse duration τ 0 / t u = γ - 1 / 2 ( t u = 1.76 k 1 / κ I sat ).

Fig. 5
Fig. 5

Dynamical evolution of an input pulse with shape ρ(s) = C exp[−(ln 2) (s/0.88)2] into a soliton on (a) the upper solution branch and (b) the lower solution branch.

Fig. 6
Fig. 6

Influence of loss on the evolution of a soliton on the lower solution branch.

Fig. 7
Fig. 7

Influence of loss on the evolution of a soliton on the upper solution branch.

Fig. 8
Fig. 8

Evolution of N = 2 and N = 3 solitons of the lower solution branch.

Fig. 9
Fig. 9

Evolution of N = 2, N = 3, and N = 4 solitons of the upper solution branch.

Fig. 10
Fig. 10

Dependence of the shortening factor τ1/τ0 of the first pulse maximum on the soliton number N for different saturation parameters γ (curve 1, γ = 0; curve 2, γ = 10−4; curve 3, γ = 10−3; curve 4, γ = 10−2).

Fig. 11
Fig. 11

Optimum fiber length z/z0 [z0 = (π/2)L0] relative to the soliton number N for different saturation parameters γ (curve 1, γ = 0; curve 2, γ = 10−4; curve 3, γ = 10−3; curve 4, γ = 10−2).

Equations (23)

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i A z + 1 2 k 1 2 η 2 A - κ f ( A 2 ) A = 0 ,
f ( A 2 ) = A 2 1 + A 2 / I sat .
Δ n = ( κ n 0 / k 1 ) f ( A 2 ) .
ξ = z / L 0 ,             s = η / t 0 ,             A = P 0 q ( ξ , s ) ,             γ = P 0 / I sat ,
- i ξ q + 1 2 2 s 2 q + q 2 q 1 + γ q 2 = 0.
q ( ξ = 0 , s = 0.88 ) = 1 / 2 q ( ξ = 0 , s = 0 ) .
q ( ξ , s , γ ) = q ( 2 ξ , s , γ ) ,
q ( ξ , s ) = [ ρ ( ξ , s ) ] 1 / 2 exp [ i φ ( ξ , s ) ] .
ρ ξ + s ( ρ φ s ) = 0 ,
1 4 ρ ρ 2 s 2 - 1 8 ρ 2 ( ρ s ) 2 + ρ 1 + γ ρ = - φ ξ + 1 2 ( φ s ) 2 .
ρ ( s ) φ s ( ξ , s ) = C ( ξ ) .
φ = - β ξ + φ 0 ,
( d ρ d s ) 2 = 8 ( β - 1 γ ) ρ 2 + ρ 8 γ 2 ln ( 1 + γ ρ ) .
β = 1 γ - ln ( 1 + γ ρ 0 ) γ 2 ρ 0 .
ρ ( s ) = ρ 0 exp ( 0 s d t { ln [ 1 + γ ρ ( t ) ] γ ρ ( t ) - ln ( 1 + γ ρ 0 ) γ ρ 0 } 1 / 2 2 2 ) .
( τ 0 ) min = 2.5 ( k 1 / κ I sat ) 1 / 2 = 2.5 ( k 1 c / Δ n sat ω ) 1 / 2 .
q ( ξ , s ) = B ( ξ ) exp { i [ φ ( ξ ) + b ( ξ ) s 2 ] } { cosh [ 1.5 s / y ( ξ ) ] } - 1 / 2 ,
1 2 ( d y d ξ ) 2 + U ( y ) = 0 , U ( y ) = 0.128 ( 1 y 2 - 1 ) + 0.29 ( γ B 0 ) 2 { π 2 4 ( 1 - y ) + y [ arccos ( γ B 0 2 y ) ] 2 - [ arccos ( γ B 2 ) ] 2 } .
γ = 1.07 B { π 2 4 - [ arccos ( γ B 2 ) ] 2 - 2 γ B 2 arccos ( γ B 2 ) [ 1 - ( γ B 2 ) ] 1 / 2 } 1 / 2 .
γ = 1 B ( 15 { ( 1 + γ B 2 ) 1 / 2 γ B ln [ ( 1 + γ B 2 ) 1 / 2 + γ B ( 1 + γ B 2 ) 1 / 2 γ B ] - 1 } - 10 ln ( 1 + γ B 2 ) ) 1 / 2 .
β ( β ) > 0 ,
i ξ q + 1 2 2 s 2 q + q 2 q 1 + γ q 2 = i Γ q .
q N ( s , ξ = 0 ) = N q ( s ) ,

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