Abstract

Soliton pulses generated by a fiber-ring laser are investigated by numerical simulation and perturbation methods. The mathematical modeling is based on the nonlinear Schrödinger equation with perturbative terms. We show that active mode locking with an amplitude modulator leads to a self-starting of stable solitonic pulses from small random noise, provided the modulation depth is small. The perturbative analysis leads to a nonlinear coupled return map for the amplitude, phase, and position of the soliton pulses circulating in the fiber-ring laser. We established the validity of this approach by comparison with the full numerical simulations. Finally, we discuss possible sources of instability that are due to resonances in the device.

© 2000 Optical Society of America

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References

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  1. S. V. Chernikov, E. M. Dianov, D. J. Richardson, and D. N. Payne, “Soliton pulse compression in dispersion-decreasing fiber,” Opt. Lett. 18, 476–478 (1993).
    [CrossRef] [PubMed]
  2. Y. K. Chen and M. C. Wu, “Monolithic colliding-pulse mode-locked quantum-well lasers,” IEEE J. Quantum Electron. 28, 2176–2185 (1992).
    [CrossRef]
  3. S. Bischoff, M. P. Sørensen, J. Mørk, S. D. Brorson, T. Franck, J. M. Nielsen, and A. Møller-Larsen, “Pulse-shaping mechanism in colliding-pulse mode-locked laser diodes,” Appl. Phys. Lett. 67, 3877–3879 (1995).
    [CrossRef]
  4. D. U. Noske, N. Pandit, and J. R. Taylor, “Source of spectral and temporal instability in soliton fiber ring lasers,” Opt. Lett. 17, 1515–1517 (1992).
    [CrossRef]
  5. J. G. Caputo, N. Flytzanis, and M. P. Sørensen, “Ring laser configuration studied by collective coordinates,” J. Opt. Soc. Am. B 12, 139–145 (1995).
    [CrossRef]
  6. A. C. Newell and J. V. Moloney, Nonlinear Optics (Addison-Wesley, Palo Alto, Calif., 1992).
  7. G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1995).
  8. E. P. Ippen, H. A. Haus, and L. Y. Liu, “Additive pulse mode locking,” J. Opt. Soc. Am. B 6, 1736–1745 (1989).
    [CrossRef]
  9. M. Nakazawa, H. Kubota, K. Kurokawa, and E. Yamada, “Femtosecond optical soliton transmission over long distances using adiabatic trapping and soliton standardization,” J. Opt. Soc. Am. B 8, 1811–1817 (1991).
    [CrossRef]
  10. T. Geisler, K. A. Shore, M. P. Soerensen, P. L. Christiansen, J. Mørk, and J. Mark, “Nonlinear fiber external cavity mode locking of erbium-doped fiber lasers,” J. Opt. Soc. Am. B 10, 1166–1174 (1993).
    [CrossRef]
  11. M. Nakazawa, K. Kurokawa, H. Kubota, K. Suzuki, and Y. Kimura, “Femtosecond erbium-doped optical fiber amplifier,” Appl. Phys. Lett. 57, 653–655 (1990).
    [CrossRef]
  12. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
  13. D. Anderson, M. Lisak, and T. Reichel, “Asymptotic propagation properties of pulses in a soliton-based optical-fiber communication system,” J. Opt. Soc. Am. B 5, 1166–1174 (1993).
  14. Y. Kodama, M. Romagnoli, and S. Wabnitz, “Stabilisation of optical solitons by an acousto-optic modulator and filter,” Electron. Lett. 30, 261–262 (1994).
    [CrossRef]
  15. C. J. Chen, P. K. A. Wai, and C. R. Menyuk, “Self-starting of passively mode-locked lasers with fast saturable absorbers,” Opt. Lett. 20, 350–352 (1995).
    [CrossRef] [PubMed]
  16. K. Tamura, E. P. Ippen, H. A. Haus, and L. E. Nelson, “77-fs pulse generation from a stretched-pulse mode-locked all-fiber ring laser,” Opt. Lett. 18, 1080–1082 (1993).
    [CrossRef] [PubMed]
  17. N. J. Smith, W. J. Firth, K. J. Blow, and K. Smith, “Suppression of soliton interactions by periodic phase modulation,” Opt. Lett. 19, 16–18 (1994).
    [CrossRef] [PubMed]
  18. H. Haus, “Mode-locked fiber ring lasers and fiber ring memories,” notes for Les Houches school “Optical solitons,” Ecole de Physique des Houches, Les Houches, France, October 1998.
  19. S. Wabnitz, “Suppression of soliton interactions by phase modulation,” Electron. Lett. 29, 1711–1713 (1993).
    [CrossRef]
  20. M. Sejka, C. Povlsen, J. H. Poulsen, Y. Shi, and O. Poulsen, “High repetition rate Q switched ring laser in Er3+ doped fiber,” Opt. Fiber Technol.: Mater., Devices Syst. 1, 167–170 (1995).
    [CrossRef]
  21. V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in non-linear media,” Sov. Phys. JETP 34, 62–69 (1972).
  22. G. Bofetta and A. R. Osborne, “Computation of the direct scattering transform for the nonlinear Schroedinger equation,” J. Comput. Phys. 102, 252–264 (1992).
    [CrossRef]
  23. F. Kh. Abdullaev and J. G. Caputo, “Propagation of an envelope soliton in a medium with spatially varying dispersion,” Phys. Rev. E 55, 6061–6071 (1997).
    [CrossRef]
  24. D. Anderson, M. Lisak, and T. Reichel, “Asymptotic propagation properties of pulses in a soliton-based optical-fiber communication system,” J. Opt. Soc. Am. B 5, 207–210 (1988).
    [CrossRef]

1997 (1)

F. Kh. Abdullaev and J. G. Caputo, “Propagation of an envelope soliton in a medium with spatially varying dispersion,” Phys. Rev. E 55, 6061–6071 (1997).
[CrossRef]

1995 (4)

S. Bischoff, M. P. Sørensen, J. Mørk, S. D. Brorson, T. Franck, J. M. Nielsen, and A. Møller-Larsen, “Pulse-shaping mechanism in colliding-pulse mode-locked laser diodes,” Appl. Phys. Lett. 67, 3877–3879 (1995).
[CrossRef]

J. G. Caputo, N. Flytzanis, and M. P. Sørensen, “Ring laser configuration studied by collective coordinates,” J. Opt. Soc. Am. B 12, 139–145 (1995).
[CrossRef]

C. J. Chen, P. K. A. Wai, and C. R. Menyuk, “Self-starting of passively mode-locked lasers with fast saturable absorbers,” Opt. Lett. 20, 350–352 (1995).
[CrossRef] [PubMed]

M. Sejka, C. Povlsen, J. H. Poulsen, Y. Shi, and O. Poulsen, “High repetition rate Q switched ring laser in Er3+ doped fiber,” Opt. Fiber Technol.: Mater., Devices Syst. 1, 167–170 (1995).
[CrossRef]

1994 (2)

N. J. Smith, W. J. Firth, K. J. Blow, and K. Smith, “Suppression of soliton interactions by periodic phase modulation,” Opt. Lett. 19, 16–18 (1994).
[CrossRef] [PubMed]

Y. Kodama, M. Romagnoli, and S. Wabnitz, “Stabilisation of optical solitons by an acousto-optic modulator and filter,” Electron. Lett. 30, 261–262 (1994).
[CrossRef]

1993 (5)

1992 (3)

Y. K. Chen and M. C. Wu, “Monolithic colliding-pulse mode-locked quantum-well lasers,” IEEE J. Quantum Electron. 28, 2176–2185 (1992).
[CrossRef]

D. U. Noske, N. Pandit, and J. R. Taylor, “Source of spectral and temporal instability in soliton fiber ring lasers,” Opt. Lett. 17, 1515–1517 (1992).
[CrossRef]

G. Bofetta and A. R. Osborne, “Computation of the direct scattering transform for the nonlinear Schroedinger equation,” J. Comput. Phys. 102, 252–264 (1992).
[CrossRef]

1991 (1)

1990 (1)

M. Nakazawa, K. Kurokawa, H. Kubota, K. Suzuki, and Y. Kimura, “Femtosecond erbium-doped optical fiber amplifier,” Appl. Phys. Lett. 57, 653–655 (1990).
[CrossRef]

1989 (1)

1988 (1)

1972 (1)

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in non-linear media,” Sov. Phys. JETP 34, 62–69 (1972).

Abdullaev, F. Kh.

F. Kh. Abdullaev and J. G. Caputo, “Propagation of an envelope soliton in a medium with spatially varying dispersion,” Phys. Rev. E 55, 6061–6071 (1997).
[CrossRef]

Anderson, D.

Bischoff, S.

S. Bischoff, M. P. Sørensen, J. Mørk, S. D. Brorson, T. Franck, J. M. Nielsen, and A. Møller-Larsen, “Pulse-shaping mechanism in colliding-pulse mode-locked laser diodes,” Appl. Phys. Lett. 67, 3877–3879 (1995).
[CrossRef]

Blow, K. J.

Bofetta, G.

G. Bofetta and A. R. Osborne, “Computation of the direct scattering transform for the nonlinear Schroedinger equation,” J. Comput. Phys. 102, 252–264 (1992).
[CrossRef]

Brorson, S. D.

S. Bischoff, M. P. Sørensen, J. Mørk, S. D. Brorson, T. Franck, J. M. Nielsen, and A. Møller-Larsen, “Pulse-shaping mechanism in colliding-pulse mode-locked laser diodes,” Appl. Phys. Lett. 67, 3877–3879 (1995).
[CrossRef]

Caputo, J. G.

F. Kh. Abdullaev and J. G. Caputo, “Propagation of an envelope soliton in a medium with spatially varying dispersion,” Phys. Rev. E 55, 6061–6071 (1997).
[CrossRef]

J. G. Caputo, N. Flytzanis, and M. P. Sørensen, “Ring laser configuration studied by collective coordinates,” J. Opt. Soc. Am. B 12, 139–145 (1995).
[CrossRef]

Chen, C. J.

Chen, Y. K.

Y. K. Chen and M. C. Wu, “Monolithic colliding-pulse mode-locked quantum-well lasers,” IEEE J. Quantum Electron. 28, 2176–2185 (1992).
[CrossRef]

Chernikov, S. V.

Christiansen, P. L.

Dianov, E. M.

Firth, W. J.

Flytzanis, N.

Franck, T.

S. Bischoff, M. P. Sørensen, J. Mørk, S. D. Brorson, T. Franck, J. M. Nielsen, and A. Møller-Larsen, “Pulse-shaping mechanism in colliding-pulse mode-locked laser diodes,” Appl. Phys. Lett. 67, 3877–3879 (1995).
[CrossRef]

Geisler, T.

Haus, H. A.

Ippen, E. P.

Kimura, Y.

M. Nakazawa, K. Kurokawa, H. Kubota, K. Suzuki, and Y. Kimura, “Femtosecond erbium-doped optical fiber amplifier,” Appl. Phys. Lett. 57, 653–655 (1990).
[CrossRef]

Kodama, Y.

Y. Kodama, M. Romagnoli, and S. Wabnitz, “Stabilisation of optical solitons by an acousto-optic modulator and filter,” Electron. Lett. 30, 261–262 (1994).
[CrossRef]

Kubota, H.

M. Nakazawa, H. Kubota, K. Kurokawa, and E. Yamada, “Femtosecond optical soliton transmission over long distances using adiabatic trapping and soliton standardization,” J. Opt. Soc. Am. B 8, 1811–1817 (1991).
[CrossRef]

M. Nakazawa, K. Kurokawa, H. Kubota, K. Suzuki, and Y. Kimura, “Femtosecond erbium-doped optical fiber amplifier,” Appl. Phys. Lett. 57, 653–655 (1990).
[CrossRef]

Kurokawa, K.

M. Nakazawa, H. Kubota, K. Kurokawa, and E. Yamada, “Femtosecond optical soliton transmission over long distances using adiabatic trapping and soliton standardization,” J. Opt. Soc. Am. B 8, 1811–1817 (1991).
[CrossRef]

M. Nakazawa, K. Kurokawa, H. Kubota, K. Suzuki, and Y. Kimura, “Femtosecond erbium-doped optical fiber amplifier,” Appl. Phys. Lett. 57, 653–655 (1990).
[CrossRef]

Lisak, M.

Liu, L. Y.

Mark, J.

Menyuk, C. R.

Møller-Larsen, A.

S. Bischoff, M. P. Sørensen, J. Mørk, S. D. Brorson, T. Franck, J. M. Nielsen, and A. Møller-Larsen, “Pulse-shaping mechanism in colliding-pulse mode-locked laser diodes,” Appl. Phys. Lett. 67, 3877–3879 (1995).
[CrossRef]

Mørk, J.

S. Bischoff, M. P. Sørensen, J. Mørk, S. D. Brorson, T. Franck, J. M. Nielsen, and A. Møller-Larsen, “Pulse-shaping mechanism in colliding-pulse mode-locked laser diodes,” Appl. Phys. Lett. 67, 3877–3879 (1995).
[CrossRef]

T. Geisler, K. A. Shore, M. P. Soerensen, P. L. Christiansen, J. Mørk, and J. Mark, “Nonlinear fiber external cavity mode locking of erbium-doped fiber lasers,” J. Opt. Soc. Am. B 10, 1166–1174 (1993).
[CrossRef]

Nakazawa, M.

M. Nakazawa, H. Kubota, K. Kurokawa, and E. Yamada, “Femtosecond optical soliton transmission over long distances using adiabatic trapping and soliton standardization,” J. Opt. Soc. Am. B 8, 1811–1817 (1991).
[CrossRef]

M. Nakazawa, K. Kurokawa, H. Kubota, K. Suzuki, and Y. Kimura, “Femtosecond erbium-doped optical fiber amplifier,” Appl. Phys. Lett. 57, 653–655 (1990).
[CrossRef]

Nelson, L. E.

Nielsen, J. M.

S. Bischoff, M. P. Sørensen, J. Mørk, S. D. Brorson, T. Franck, J. M. Nielsen, and A. Møller-Larsen, “Pulse-shaping mechanism in colliding-pulse mode-locked laser diodes,” Appl. Phys. Lett. 67, 3877–3879 (1995).
[CrossRef]

Noske, D. U.

Osborne, A. R.

G. Bofetta and A. R. Osborne, “Computation of the direct scattering transform for the nonlinear Schroedinger equation,” J. Comput. Phys. 102, 252–264 (1992).
[CrossRef]

Pandit, N.

Payne, D. N.

Poulsen, J. H.

M. Sejka, C. Povlsen, J. H. Poulsen, Y. Shi, and O. Poulsen, “High repetition rate Q switched ring laser in Er3+ doped fiber,” Opt. Fiber Technol.: Mater., Devices Syst. 1, 167–170 (1995).
[CrossRef]

Poulsen, O.

M. Sejka, C. Povlsen, J. H. Poulsen, Y. Shi, and O. Poulsen, “High repetition rate Q switched ring laser in Er3+ doped fiber,” Opt. Fiber Technol.: Mater., Devices Syst. 1, 167–170 (1995).
[CrossRef]

Povlsen, C.

M. Sejka, C. Povlsen, J. H. Poulsen, Y. Shi, and O. Poulsen, “High repetition rate Q switched ring laser in Er3+ doped fiber,” Opt. Fiber Technol.: Mater., Devices Syst. 1, 167–170 (1995).
[CrossRef]

Reichel, T.

Richardson, D. J.

Romagnoli, M.

Y. Kodama, M. Romagnoli, and S. Wabnitz, “Stabilisation of optical solitons by an acousto-optic modulator and filter,” Electron. Lett. 30, 261–262 (1994).
[CrossRef]

Sejka, M.

M. Sejka, C. Povlsen, J. H. Poulsen, Y. Shi, and O. Poulsen, “High repetition rate Q switched ring laser in Er3+ doped fiber,” Opt. Fiber Technol.: Mater., Devices Syst. 1, 167–170 (1995).
[CrossRef]

Shabat, A. B.

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in non-linear media,” Sov. Phys. JETP 34, 62–69 (1972).

Shi, Y.

M. Sejka, C. Povlsen, J. H. Poulsen, Y. Shi, and O. Poulsen, “High repetition rate Q switched ring laser in Er3+ doped fiber,” Opt. Fiber Technol.: Mater., Devices Syst. 1, 167–170 (1995).
[CrossRef]

Shore, K. A.

Smith, K.

Smith, N. J.

Soerensen, M. P.

Sørensen, M. P.

S. Bischoff, M. P. Sørensen, J. Mørk, S. D. Brorson, T. Franck, J. M. Nielsen, and A. Møller-Larsen, “Pulse-shaping mechanism in colliding-pulse mode-locked laser diodes,” Appl. Phys. Lett. 67, 3877–3879 (1995).
[CrossRef]

J. G. Caputo, N. Flytzanis, and M. P. Sørensen, “Ring laser configuration studied by collective coordinates,” J. Opt. Soc. Am. B 12, 139–145 (1995).
[CrossRef]

Suzuki, K.

M. Nakazawa, K. Kurokawa, H. Kubota, K. Suzuki, and Y. Kimura, “Femtosecond erbium-doped optical fiber amplifier,” Appl. Phys. Lett. 57, 653–655 (1990).
[CrossRef]

Tamura, K.

Taylor, J. R.

Wabnitz, S.

Y. Kodama, M. Romagnoli, and S. Wabnitz, “Stabilisation of optical solitons by an acousto-optic modulator and filter,” Electron. Lett. 30, 261–262 (1994).
[CrossRef]

S. Wabnitz, “Suppression of soliton interactions by phase modulation,” Electron. Lett. 29, 1711–1713 (1993).
[CrossRef]

Wai, P. K. A.

Wu, M. C.

Y. K. Chen and M. C. Wu, “Monolithic colliding-pulse mode-locked quantum-well lasers,” IEEE J. Quantum Electron. 28, 2176–2185 (1992).
[CrossRef]

Yamada, E.

Zakharov, V. E.

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in non-linear media,” Sov. Phys. JETP 34, 62–69 (1972).

Appl. Phys. Lett. (2)

S. Bischoff, M. P. Sørensen, J. Mørk, S. D. Brorson, T. Franck, J. M. Nielsen, and A. Møller-Larsen, “Pulse-shaping mechanism in colliding-pulse mode-locked laser diodes,” Appl. Phys. Lett. 67, 3877–3879 (1995).
[CrossRef]

M. Nakazawa, K. Kurokawa, H. Kubota, K. Suzuki, and Y. Kimura, “Femtosecond erbium-doped optical fiber amplifier,” Appl. Phys. Lett. 57, 653–655 (1990).
[CrossRef]

Electron. Lett. (2)

S. Wabnitz, “Suppression of soliton interactions by phase modulation,” Electron. Lett. 29, 1711–1713 (1993).
[CrossRef]

Y. Kodama, M. Romagnoli, and S. Wabnitz, “Stabilisation of optical solitons by an acousto-optic modulator and filter,” Electron. Lett. 30, 261–262 (1994).
[CrossRef]

IEEE J. Quantum Electron. (1)

Y. K. Chen and M. C. Wu, “Monolithic colliding-pulse mode-locked quantum-well lasers,” IEEE J. Quantum Electron. 28, 2176–2185 (1992).
[CrossRef]

J. Comput. Phys. (1)

G. Bofetta and A. R. Osborne, “Computation of the direct scattering transform for the nonlinear Schroedinger equation,” J. Comput. Phys. 102, 252–264 (1992).
[CrossRef]

J. Opt. Soc. Am. B (6)

Opt. Fiber Technol.: Mater., Devices Syst. (1)

M. Sejka, C. Povlsen, J. H. Poulsen, Y. Shi, and O. Poulsen, “High repetition rate Q switched ring laser in Er3+ doped fiber,” Opt. Fiber Technol.: Mater., Devices Syst. 1, 167–170 (1995).
[CrossRef]

Opt. Lett. (5)

Phys. Rev. E (1)

F. Kh. Abdullaev and J. G. Caputo, “Propagation of an envelope soliton in a medium with spatially varying dispersion,” Phys. Rev. E 55, 6061–6071 (1997).
[CrossRef]

Sov. Phys. JETP (1)

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in non-linear media,” Sov. Phys. JETP 34, 62–69 (1972).

Other (4)

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).

H. Haus, “Mode-locked fiber ring lasers and fiber ring memories,” notes for Les Houches school “Optical solitons,” Ecole de Physique des Houches, Les Houches, France, October 1998.

A. C. Newell and J. V. Moloney, Nonlinear Optics (Addison-Wesley, Palo Alto, Calif., 1992).

G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1995).

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Figures (7)

Fig. 1
Fig. 1

Schematic drawing of the fiber-ring laser with the amplitude modulator.

Fig. 2
Fig. 2

Failing self-starting without the amplitude modulator. The real and imaginary parts of the initial condition are equally distributed random numbers between -10-2 and +10-2. The gain g0=4; we assumed no detuning δm=0.0 and an output coupler γc=0.01. The other parameters are given in Table 1.

Fig. 3
Fig. 3

Self-starting from a random-noise initial condition with an amplitude modulator, α0=0.98, αm=0.02, ωm=2π/25=0.2513, and δm=0.0; the other parameters are given in Table 1.

Fig. 4
Fig. 4

Self-starting from noise with an amplitude modulator and detuning δm=0.04. Other parameter values are as in Fig. 3.

Fig. 5
Fig. 5

Failing self-starting with phase modulation and a random-noise initial condition. Parameters are as in Fig. 3.

Fig. 6
Fig. 6

Amplitude η as function of the modulation depth k for zero detuning and three values of the gain g0=2 (bottom curve), g0=4 (middle curve), and g0=6 (top curve). Solid curves denote numerical simulation, and dashed curves denote the return map in Eq. (3.21). The middle plate shows the soliton content obtained from the numerical solution, and the bottom plate gives the chirp. Parameters are as in Fig. 2.

Fig. 7
Fig. 7

Same as in Fig. 6 except g0=5 (bottom curve), g0=10 (middle curve), g0=15 (top curve), γc=0.05, la=0.4, and lp=0.1.

Tables (1)

Tables Icon

Table 1 Fixed Parameter Values Used Throughout in the Numerical Computations

Equations (48)

Equations on this page are rendered with MathJax. Learn more.

AZ+β1AT+i2β22AT2-iγ|A|2A
=-α2A+G01+P/PsatA.
P=Frep-+|A(Z, T)|2dT,
IoIi=α1+α2sin[Γmsin(ΩmT)],
AoAi=α0+αm sin(ΩmT+ϕm),
AoAi=1-Γc.
t=kt(T-β1Z),z=kz Z,A(Z, T)=ku(z, t),
iuz+2ut2+2|u|2u=-iΓu+ig01+p/psatu=R,
kz=-12β2kt2,k2=2kz/γ,
Γ=α/(2kz),g0=G0/kz,
psat=kt Psat/(Frep k2).
p=-+|u(z, t)|2dt.
uoui=α0+αm sin(ωmt+nδm+ϕm),
ωm=Ωmkt,δm=β1Ωm(La+Lb)mod2π.
u(z, t)=η sech(ηθ)exp(iξθ+iσ),
θ=t-2ξ z-t0,
σ=(η2+ξ2)z-σ0.
L=-i2(u*uz-uz*u)-utut*+u2u*2dt.
Ly-ddzLyz=-Ru*ydt+cc,
iuz+2ut2+2|u|2u=-iχδ(z-zm)u,
ilogu(zm+)u(zm-)+zm-zm+dzuttu+2|u|2=-iχ,
u(zm+)=u(zm-)exp(-χ)
R1=-iχmδ(z-zm)u-iχocδ(z-zoc)u
χm=-log(1-αm)-αmα0sin(ωmt+ϕm),
χoc=-log(1-Γc).
dηdz=-2Γη+2g0η1+(p/psat),
dξdz=0,
dt0dz=0,
dσ0dz=2zηdηdz,
dηdz=-ωmπαmα0sin(ψ)sinh(πωm/2η)δ(z-zm)+2 log(α0)ηδ(z-zm),
dξdz=0,
dt0dz=2παmα0cos(ψ)2η sinh(πωm/2η)×1-πωm2ηcothπωm2ηδ(z-zm),
dσ0dz=2zηdηdz+παmα0cos(ψ)2η sinh(πωm/2η)×1-πωm2ηcothπωm2ηξδ(z-zm),
dηdz=2log(1-Γc)ηδ(z-zoc).
η2=ηn exp2g0la1+(2ηn/psat)-2Γla,
η3=η2[1+2log(α0)]-ωmπαmα0sin(ψn)sinh(πωm/2η2),
t0n+1=t0n+παmα0cos(ψn)η2 sinh(πωm/2η)×1-πωm2η2cothπωm2η2,
η4=η3 exp(-2Γlp),
ηn+1=[1+2log(1-Γc)]η4.
η2=ηn exp2g0la1+(2ηn/psat)-2Γ(la+lp),
η3=n2[1+2log(α0)]-ωmπαmα0sin(ψn)sinh(πωm/2η2),
ψn+1=ψn+ωmπαmα0cos(ψn)η2 sinh(πωm/2η)×1-πωm2η2cothπωm2η2,
ηn+1=[1+2log(1-Γc)]η3.
η*=psat22g0la2Γ(la+lp)-log {[1+2log(1-Γc)][1+2log(α0)+2(αm/α0)]}-1.
uoui=exp[iαm sin(ΩmT+ϕm)].
Δsol=4Im(ζ)-+|u|2dt.
iuz+2ut2+2|u|2u=-ig(z)u,
la*=1+1+4A2lp2A2,

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