Abstract

A theoretical model is developed for the pulse dynamics in a fiber laser mode locked by a saturable Bragg reflector and operating in regimes beyond the scope of the master mode-locking equation. An asymptotically valid mode-locked evolution equation is derived, which includes a heuristic model for the saturable Bragg reflector dynamics. The model employed allows, for the first time to our knowledge, direct comparison (with no free parameters) of the theoretical predictions of the pulse spectral and temporal profiles with experimental results in both the normal and anomalous dispersion regimes. Extensive numerical simulations of the governing evolution equation, an averaged equation, and analytical solutions are found to be in excellent agreement with experimental results.

© 1997 Optical Society of America

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    [CrossRef]
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1997

J. C. Bronski and J. N. Kutz, “Asymptotic behavior of the nonlinear Schrödinger equation withrapidly-varying, mean-zero dispersion,” Physica D 108, 315–329 (1997).
[CrossRef]

J. C. Bronski and J. N. Kutz, “Guiding-center pulse dynamics in nonreturn-to-zero (return-to-zero)communications system with mean-zero dispersion,” J. Opt. Soc. Am. B 14, 903–911 (1997).
[CrossRef]

1996

1995

1994

H. A. Haus, E. P. Ippen, and K. Tamura, “Additive pulse modelocking in fiber lasers,” IEEE J. Quantum Electron. 30, 200–208 (1994).
[CrossRef]

T. Y. Hou, J. S. Lowengrub, and M. J. Shelley, “Removing the stiffness from interfacial flows with surface tension,” J. Comput. Phys. 114, 312–338 (1994).
[CrossRef]

F. X. Kärtner, D. Kopf, and U. Keller, “Solitary pulse stabilization and shortening in actively modelockedlasers,” J. Opt. Soc. Am. B 12, 486–496 (1994).
[CrossRef]

C.-J. Chen, P. K. A. Wai, and C. R. Menyuk, “Stability of passively mode-locked fiber lasers with fast saturableabsorption,” Opt. Lett. 19, 198–200 (1994).
[CrossRef]

1993

M. E. Fermann, M. J. Andrejco, Y. Silverberg, and M. L. Stock, “Passive modelocking by using nonlinear polarization evolution in apolarizing-maintaining erbium-doped fiber,” Opt. Lett. 18, 894–896 (1993).
[CrossRef]

P. F. Curley, C. Spielmann, T. Brabec, E. Winter, and F. Krausz, “Periodic pulse evolution in solitary lasers,” J. Opt. Soc. Am. B 10, 1025–1028 (1993).
[CrossRef]

H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” J. Lightwave Technol. 29, 983–996 (1993).

E. A. DeSouza, C. E. Soccolich, W. Pleibel, R. H. Stolen, M. N. Islam, J. R. Simpson, and D. J. DiGiovanni, “Saturable absorber modelocked polarization maintaining erbium-dopedfiber laser,” Electron. Lett. 29, 447–449 (1993).
[CrossRef]

1992

M. Hofer, M. H. Ober, F. Haberl, and M. E. Fermann, “Characterization of ultrashort pulse formation in passively modelockedfiber lasers,” IEEE J. Quantum Electron. 28, 720–728 (1992).
[CrossRef]

F. Krausz, M. E. Fermann, T. Brabec, P. F. Curley, M. Hofer, M. H. Ober, C. Speilmann, E. Wintner, and A. J. Schmit, “Femtosecond solid-state lasers,” IEEE J. Quantum Electron. 28, 2097–2122 (1992).
[CrossRef]

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse mode-locking and Kerr lens modelocking,” IEEE J. Quantum Electron. 28, 2086–2096 (1992).
[CrossRef]

M. L. Dennis and I. N. Duling III, “High repetition rate figure eight laser with extracavity feedback,” Electron. Lett. 28, 1894–1896 (1992).
[CrossRef]

K. Tamura, H. A. Haus, and E. P. Ippen, “Self-starting additive pulse modelocked erbium fiber ring laser,” Electron. Lett. 28, 2226–2228 (1992).
[CrossRef]

T. Brabec, C. Spielmann, and F. Krausz, “Limits of pulse shortening in solitary lasers,” Opt. Lett. 17, 748–750 (1992).
[CrossRef] [PubMed]

1991

1989

D. S. Chemla, W. H. Knox, D. A. B. Miller, S. Schmitt-Rink, J. B. Stark, and R. Zimmermann, “The excitonic optical Stark effect in semiconductor quantum wells probedwith femtosecond optical pulses,” J. Lumin. 44, 233–246 (1989).
[CrossRef]

1984

T. R. Taha and M. J. Ablowitz, “Analytical and numerical aspects of certain nonlinear evolution equations.II. Numerical, nonlinear Schrödinger equation,” J. Comput. Phys. 55, 203–230 (1984).
[CrossRef]

1978

1977

N. R. Pereira and L. Stenflo, “Nonlinear Schrödinger equation including growth and damping,” Phys. Fluids 20, 1733–1734 (1977).
[CrossRef]

Ablowitz, M. J.

T. R. Taha and M. J. Ablowitz, “Analytical and numerical aspects of certain nonlinear evolution equations.II. Numerical, nonlinear Schrödinger equation,” J. Comput. Phys. 55, 203–230 (1984).
[CrossRef]

Andrejco, M. J.

Bergman, K.

Blow, K. J.

Bossalini, L.

Brabec, T.

Bronski, J. C.

J. C. Bronski and J. N. Kutz, “Guiding-center pulse dynamics in nonreturn-to-zero (return-to-zero)communications system with mean-zero dispersion,” J. Opt. Soc. Am. B 14, 903–911 (1997).
[CrossRef]

J. C. Bronski and J. N. Kutz, “Asymptotic behavior of the nonlinear Schrödinger equation withrapidly-varying, mean-zero dispersion,” Physica D 108, 315–329 (1997).
[CrossRef]

Brovelli, L. R.

F. X. Kartner, L. R. Brovelli, D. Kopf, M. Kamp, I. Calasso, and U. Keller, “Control of solid-state laser dynamics by semiconductor devices,” Opt. Eng. 34, 2024–2036 (1995).
[CrossRef]

Calasso, I.

F. X. Kartner, L. R. Brovelli, D. Kopf, M. Kamp, I. Calasso, and U. Keller, “Control of solid-state laser dynamics by semiconductor devices,” Opt. Eng. 34, 2024–2036 (1995).
[CrossRef]

Chemla, D. S.

D. S. Chemla, W. H. Knox, D. A. B. Miller, S. Schmitt-Rink, J. B. Stark, and R. Zimmermann, “The excitonic optical Stark effect in semiconductor quantum wells probedwith femtosecond optical pulses,” J. Lumin. 44, 233–246 (1989).
[CrossRef]

Chen, C.-J.

Collings, B. C.

Cunningham, J. E.

Curley, P. F.

P. F. Curley, C. Spielmann, T. Brabec, E. Winter, and F. Krausz, “Periodic pulse evolution in solitary lasers,” J. Opt. Soc. Am. B 10, 1025–1028 (1993).
[CrossRef]

F. Krausz, M. E. Fermann, T. Brabec, P. F. Curley, M. Hofer, M. H. Ober, C. Speilmann, E. Wintner, and A. J. Schmit, “Femtosecond solid-state lasers,” IEEE J. Quantum Electron. 28, 2097–2122 (1992).
[CrossRef]

de Souza, E. A.

Dennis, M. L.

M. L. Dennis and I. N. Duling III, “High repetition rate figure eight laser with extracavity feedback,” Electron. Lett. 28, 1894–1896 (1992).
[CrossRef]

DeSouza, E. A.

E. A. DeSouza, C. E. Soccolich, W. Pleibel, R. H. Stolen, M. N. Islam, J. R. Simpson, and D. J. DiGiovanni, “Saturable absorber modelocked polarization maintaining erbium-dopedfiber laser,” Electron. Lett. 29, 447–449 (1993).
[CrossRef]

DiGiovanni, D. J.

E. A. DeSouza, C. E. Soccolich, W. Pleibel, R. H. Stolen, M. N. Islam, J. R. Simpson, and D. J. DiGiovanni, “Saturable absorber modelocked polarization maintaining erbium-dopedfiber laser,” Electron. Lett. 29, 447–449 (1993).
[CrossRef]

Doran, N. J.

Duling III, I. N.

M. L. Dennis and I. N. Duling III, “High repetition rate figure eight laser with extracavity feedback,” Electron. Lett. 28, 1894–1896 (1992).
[CrossRef]

I. N. Duling III, “Subpicosecond all-fiber erbium laser,” Electron. Lett. 27, 544–545 (1991).
[CrossRef]

Fermann, M. E.

M. E. Fermann, M. J. Andrejco, Y. Silverberg, and M. L. Stock, “Passive modelocking by using nonlinear polarization evolution in apolarizing-maintaining erbium-doped fiber,” Opt. Lett. 18, 894–896 (1993).
[CrossRef]

F. Krausz, M. E. Fermann, T. Brabec, P. F. Curley, M. Hofer, M. H. Ober, C. Speilmann, E. Wintner, and A. J. Schmit, “Femtosecond solid-state lasers,” IEEE J. Quantum Electron. 28, 2097–2122 (1992).
[CrossRef]

M. Hofer, M. H. Ober, F. Haberl, and M. E. Fermann, “Characterization of ultrashort pulse formation in passively modelockedfiber lasers,” IEEE J. Quantum Electron. 28, 720–728 (1992).
[CrossRef]

Fleck, J. A.

Fleit, M. D.

Fontana, F.

Franco, P.

Fujimoto, J. G.

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse mode-locking and Kerr lens modelocking,” IEEE J. Quantum Electron. 28, 2086–2096 (1992).
[CrossRef]

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Structures for additive pulse mode locking,” J. Opt. Soc. Am. B 8, 2068–2076 (1991).
[CrossRef]

Haberl, F.

M. Hofer, M. H. Ober, F. Haberl, and M. E. Fermann, “Characterization of ultrashort pulse formation in passively modelockedfiber lasers,” IEEE J. Quantum Electron. 28, 720–728 (1992).
[CrossRef]

Hasegawa, A.

Haus, H. A.

H. A. Haus, E. P. Ippen, and K. Tamura, “Additive pulse modelocking in fiber lasers,” IEEE J. Quantum Electron. 30, 200–208 (1994).
[CrossRef]

H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” J. Lightwave Technol. 29, 983–996 (1993).

K. Tamura, H. A. Haus, and E. P. Ippen, “Self-starting additive pulse modelocked erbium fiber ring laser,” Electron. Lett. 28, 2226–2228 (1992).
[CrossRef]

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse mode-locking and Kerr lens modelocking,” IEEE J. Quantum Electron. 28, 2086–2096 (1992).
[CrossRef]

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Structures for additive pulse mode locking,” J. Opt. Soc. Am. B 8, 2068–2076 (1991).
[CrossRef]

Hofer, M.

M. Hofer, M. H. Ober, F. Haberl, and M. E. Fermann, “Characterization of ultrashort pulse formation in passively modelockedfiber lasers,” IEEE J. Quantum Electron. 28, 720–728 (1992).
[CrossRef]

F. Krausz, M. E. Fermann, T. Brabec, P. F. Curley, M. Hofer, M. H. Ober, C. Speilmann, E. Wintner, and A. J. Schmit, “Femtosecond solid-state lasers,” IEEE J. Quantum Electron. 28, 2097–2122 (1992).
[CrossRef]

Hou, T. Y.

T. Y. Hou, J. S. Lowengrub, and M. J. Shelley, “Removing the stiffness from interfacial flows with surface tension,” J. Comput. Phys. 114, 312–338 (1994).
[CrossRef]

Ippen, E. P.

H. A. Haus, E. P. Ippen, and K. Tamura, “Additive pulse modelocking in fiber lasers,” IEEE J. Quantum Electron. 30, 200–208 (1994).
[CrossRef]

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse mode-locking and Kerr lens modelocking,” IEEE J. Quantum Electron. 28, 2086–2096 (1992).
[CrossRef]

K. Tamura, H. A. Haus, and E. P. Ippen, “Self-starting additive pulse modelocked erbium fiber ring laser,” Electron. Lett. 28, 2226–2228 (1992).
[CrossRef]

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Structures for additive pulse mode locking,” J. Opt. Soc. Am. B 8, 2068–2076 (1991).
[CrossRef]

Islam, M. N.

E. A. DeSouza, C. E. Soccolich, W. Pleibel, R. H. Stolen, M. N. Islam, J. R. Simpson, and D. J. DiGiovanni, “Saturable absorber modelocked polarization maintaining erbium-dopedfiber laser,” Electron. Lett. 29, 447–449 (1993).
[CrossRef]

Jan, W. Y.

Kamp, M.

F. X. Kartner, L. R. Brovelli, D. Kopf, M. Kamp, I. Calasso, and U. Keller, “Control of solid-state laser dynamics by semiconductor devices,” Opt. Eng. 34, 2024–2036 (1995).
[CrossRef]

Kartner, F. X.

F. X. Kartner, L. R. Brovelli, D. Kopf, M. Kamp, I. Calasso, and U. Keller, “Control of solid-state laser dynamics by semiconductor devices,” Opt. Eng. 34, 2024–2036 (1995).
[CrossRef]

Kärtner, F. X.

Keller, U.

F. X. Kartner, L. R. Brovelli, D. Kopf, M. Kamp, I. Calasso, and U. Keller, “Control of solid-state laser dynamics by semiconductor devices,” Opt. Eng. 34, 2024–2036 (1995).
[CrossRef]

F. X. Kärtner, D. Kopf, and U. Keller, “Solitary pulse stabilization and shortening in actively modelockedlasers,” J. Opt. Soc. Am. B 12, 486–496 (1994).
[CrossRef]

Kelly, S. M. J.

Knox, W. H.

Kodama, Y.

Kopf, D.

F. X. Kartner, L. R. Brovelli, D. Kopf, M. Kamp, I. Calasso, and U. Keller, “Control of solid-state laser dynamics by semiconductor devices,” Opt. Eng. 34, 2024–2036 (1995).
[CrossRef]

F. X. Kärtner, D. Kopf, and U. Keller, “Solitary pulse stabilization and shortening in actively modelockedlasers,” J. Opt. Soc. Am. B 12, 486–496 (1994).
[CrossRef]

Krausz, F.

Kutz, J. N.

J. C. Bronski and J. N. Kutz, “Guiding-center pulse dynamics in nonreturn-to-zero (return-to-zero)communications system with mean-zero dispersion,” J. Opt. Soc. Am. B 14, 903–911 (1997).
[CrossRef]

J. C. Bronski and J. N. Kutz, “Asymptotic behavior of the nonlinear Schrödinger equation withrapidly-varying, mean-zero dispersion,” Physica D 108, 315–329 (1997).
[CrossRef]

Laming, R. I.

D. J. Richardson, R. I. Laming, D. N. Payne, V. J. Matsas, and M. W. Phillips, “Self-starting, passively modelocked erbium fiber laser based on theamplifying Sagnac switch,” Electron. Lett. 27, 542–544 (1991).
[CrossRef]

Lowengrub, J. S.

T. Y. Hou, J. S. Lowengrub, and M. J. Shelley, “Removing the stiffness from interfacial flows with surface tension,” J. Comput. Phys. 114, 312–338 (1994).
[CrossRef]

Matsas, V. J.

D. J. Richardson, R. I. Laming, D. N. Payne, V. J. Matsas, and M. W. Phillips, “Self-starting, passively modelocked erbium fiber laser based on theamplifying Sagnac switch,” Electron. Lett. 27, 542–544 (1991).
[CrossRef]

Mecozzi, A.

H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” J. Lightwave Technol. 29, 983–996 (1993).

Menyuk, C. R.

Midrio, M.

Miller, D. A. B.

D. S. Chemla, W. H. Knox, D. A. B. Miller, S. Schmitt-Rink, J. B. Stark, and R. Zimmermann, “The excitonic optical Stark effect in semiconductor quantum wells probedwith femtosecond optical pulses,” J. Lumin. 44, 233–246 (1989).
[CrossRef]

Ober, M. H.

M. Hofer, M. H. Ober, F. Haberl, and M. E. Fermann, “Characterization of ultrashort pulse formation in passively modelockedfiber lasers,” IEEE J. Quantum Electron. 28, 720–728 (1992).
[CrossRef]

F. Krausz, M. E. Fermann, T. Brabec, P. F. Curley, M. Hofer, M. H. Ober, C. Speilmann, E. Wintner, and A. J. Schmit, “Femtosecond solid-state lasers,” IEEE J. Quantum Electron. 28, 2097–2122 (1992).
[CrossRef]

Pathak, R.

Payne, D. N.

D. J. Richardson, R. I. Laming, D. N. Payne, V. J. Matsas, and M. W. Phillips, “Self-starting, passively modelocked erbium fiber laser based on theamplifying Sagnac switch,” Electron. Lett. 27, 542–544 (1991).
[CrossRef]

Pereira, N. R.

N. R. Pereira and L. Stenflo, “Nonlinear Schrödinger equation including growth and damping,” Phys. Fluids 20, 1733–1734 (1977).
[CrossRef]

Phillips, M. W.

D. J. Richardson, R. I. Laming, D. N. Payne, V. J. Matsas, and M. W. Phillips, “Self-starting, passively modelocked erbium fiber laser based on theamplifying Sagnac switch,” Electron. Lett. 27, 542–544 (1991).
[CrossRef]

Pleibel, W.

E. A. DeSouza, C. E. Soccolich, W. Pleibel, R. H. Stolen, M. N. Islam, J. R. Simpson, and D. J. DiGiovanni, “Saturable absorber modelocked polarization maintaining erbium-dopedfiber laser,” Electron. Lett. 29, 447–449 (1993).
[CrossRef]

Richardson, D. J.

D. J. Richardson, R. I. Laming, D. N. Payne, V. J. Matsas, and M. W. Phillips, “Self-starting, passively modelocked erbium fiber laser based on theamplifying Sagnac switch,” Electron. Lett. 27, 542–544 (1991).
[CrossRef]

Romagnoli, M.

Schmit, A. J.

F. Krausz, M. E. Fermann, T. Brabec, P. F. Curley, M. Hofer, M. H. Ober, C. Speilmann, E. Wintner, and A. J. Schmit, “Femtosecond solid-state lasers,” IEEE J. Quantum Electron. 28, 2097–2122 (1992).
[CrossRef]

Schmitt-Rink, S.

D. S. Chemla, W. H. Knox, D. A. B. Miller, S. Schmitt-Rink, J. B. Stark, and R. Zimmermann, “The excitonic optical Stark effect in semiconductor quantum wells probedwith femtosecond optical pulses,” J. Lumin. 44, 233–246 (1989).
[CrossRef]

Shelley, M. J.

T. Y. Hou, J. S. Lowengrub, and M. J. Shelley, “Removing the stiffness from interfacial flows with surface tension,” J. Comput. Phys. 114, 312–338 (1994).
[CrossRef]

Silverberg, Y.

Simpson, J. R.

E. A. DeSouza, C. E. Soccolich, W. Pleibel, R. H. Stolen, M. N. Islam, J. R. Simpson, and D. J. DiGiovanni, “Saturable absorber modelocked polarization maintaining erbium-dopedfiber laser,” Electron. Lett. 29, 447–449 (1993).
[CrossRef]

Smith, K.

Soccolich, C. E.

E. A. DeSouza, C. E. Soccolich, W. Pleibel, R. H. Stolen, M. N. Islam, J. R. Simpson, and D. J. DiGiovanni, “Saturable absorber modelocked polarization maintaining erbium-dopedfiber laser,” Electron. Lett. 29, 447–449 (1993).
[CrossRef]

Speilmann, C.

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Figures (10)

Fig. 1
Fig. 1

Schematic of mode-locking fiber laser, which includes a 1% output coupler, SBR, and a gain-saturated fiber segment.

Fig. 2
Fig. 2

SBR structure and its associated reflectivity spectrum.

Fig. 3
Fig. 3

(a) Change in reflectivity σf-Δf corresponding to the fast response. (b) σs-Δs gives the change in reflectivity that is due to the slow saturated response with relaxation. Note that Q- corresponds to a 1-ps hyperbolic-secant pulse.

Fig. 4
Fig. 4

Comparison of the experimental pump and probe measurements of the SBR at 1.55 µm with the jump condition of Eq. (7), which includes a fast response and slow saturation response with relaxation. Here, an 400-fs hyperbolic-secant pulse is assumed for Q-, and σf=σs=0.65σ with Td=14.

Fig. 5
Fig. 5

Experimental setup of mode-locking laser cavity with a 1% output coupler, SBR, and average dispersion in the anomalous regime. The 980-nm pump is coupled through a wavelength division multiplexer (WDM) into the laser cavity, which consists of three fiber segments: a single-mode fiber (SMF), an erbium–ytterbium fiber (Er/Yb), and finally a dispersion-shifted fiber (DSF).

Fig. 6
Fig. 6

Numerical simulation of the pulse evolution as governed by Eqs. (1), (4), and (11) in both (a) the time and (b) the frequency domain for a fiber cavity with an average normal dispersion value of D¯=-38.1 ps/(km nm). In (a) the pulse is seen to mode lock to a 13.6 ps chirped pulse whose spectral evolution is given by (b).

Fig. 7
Fig. 7

Comparison of the experimental results with D¯ -38 ps/(km nm) and the full evolution equations [Eqs. (1), (4), and (11)], the averaged equation [Eq. (13)], and the approximate solution with no slow response [Eqs. (14)–(19)]. Note the remarkable agreement between all three models and the experimental results. The 13.5-ps pulse width is consistent with experimental observation.

Fig. 8
Fig. 8

Numerical simulation of the pulse evolution as governed by Eqs. (1), (4), and (11) in both (a) the time and (b) the frequency domain for a fiber cavity with an average anomalous dispersion value of D¯12 ps/(km nm). In (a) the laser is seen to mode lock to an 420-fs chirped pulse with a time–bandwidth product of 0.4 and whose spectral evolution is given by (b).

Fig. 9
Fig. 9

Comparison of the experimental autocorrelation measurements (scaled with the assumption that the pulse shape is a hyperbolic secant) and output spectrum with the full evolution equations [Eqs. (1), (4), and (11)], the averaged equation [Eq. 13)], and the approximate analytic solution with no slow response [Eqs. (14)–(19)].

Fig. 10
Fig. 10

Typical evolution of a pulse governed by full evolution equations (1), (4), and (11) in the anomalous dispersion regime D¯12 ps/(km nm). Note that for the estimated experimental parameter values the pulse begins to break up into a group of multiple pulses. In this simulation we have taken σs=0 and σf=2.6%.

Equations (31)

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i QZ+D22QT2+α|Q|2Q+iΓQ
-iG(Z)1+τ 2T2Q=0.
D=Zcavλ02D¯2πcT02
α=2πn2|E0|2Zcavλ0Aeff
τ=1Ω2T02
Γ=γ˜Zcav,
G(Z)=2G01+Q(Z)2/Esat,
Q+=1-σl-σf1-|Q-|2|Q-|max2-σs1--T|Q-|2dTQ-2×exp-H(T-Tmax) T-TmaxTdQ-=f(|Q|)Q-,
Δf=1-f(|Q|, σs=σl=0)=σf tanh2 wT.
Δs=1-f(|Q|, σf=σl=0)=σs2×[2-(1+tanh wT)exp[-H(T)T/Td)].
f(|Q|)=1-σl-σf tanh2 wT-σs2{2-(1+tanh wT)exp[-H(T)T/Td]}.
QZ=βZcavQ.
exp(βZcav)=f(|Q|).
βZcav-σl-σf1-|Q|2|Q|max2-σs1--T|Q|2dTQ2×exp-H(T-Tmax) T-TmaxTd,
Q+=RQ-,
βZcav(1-R),
i QZ+D2-iτG(z) 2QT2+i[γ-G(Z)]Q+α-i σf|Q|max2|Q|2Q-iσs -T|Q|2dTQ2
×exp-H(T-Tmax) T-TmaxTdQ=0,
Q(Z, T)=η[sech wT]1+iA exp[iϕ(Z)],
dϕdZ=12Dw2(1-A2)+2τG0w2A,
η2=w22α[D(2-A2)+6τG0A],
Dw2A-[τw2(1-A2)+1]G0+γ=0,
3Dw2A-2τG0w2(2-A2)+2σf=0.
A2(σf+B)+A DτG0σf+32B-(σf+2B)=0,
A±=1σf+B-D2τG0σf+32B±D24τ2G02σf+32B2+(σf+B)(σf+2B)1/2.
w±2=2σf2τG0(2-A±2)-3DA±,
Q(Z, T)=η[sech w(T-vZ)]1+iA exp[iϕ(Z)].
v(1+iA)=σs2w[1+coth w(T-vZ)].
vσsw.
σ¯s=σs2(t1+t2)-t1t22-(1+tanh wt)exp-H(t)tTddt0.65σs.
Q(0, T)=η0 sech(1.76w0T).

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