Abstract

The propagation of an optical soliton is studied in a fiber with randomly varying core diameter or random amplification along the fiber. The adiabatic dynamics of the soliton and radiative processes are investigated by a perturbation method based on the inverse scattering transform. The mean emitted power and the damping rate of the soliton are calculated. The interaction of solitons in random media is investigated by the Karpman–Solov’ev perturbation theory. Numerical simulations of the full nonlinear Schrödinger equation with multiplicative white- and colored-noise perturbations are performed for initial conditions corresponding to a single soliton and to two interacting solitons. The results obtained are in good agreement with the analytical estimates for white noise in the initial stage of the propagation. The numerical simulations reveal a new phenomenon in which single solitons disintegrate or split under white-noise perturbation but stabilize under colored-noise action. Finally, the existence of a bound state of two interacting solitons is observed in numerical simulations in media with colored-noise perturbation.

© 1998 Optical Society of America

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References

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  1. A. Hasegawa and Yu. Kodama, Solitons in Optical Communications (Clarendon, Oxford, 1995).
  2. J. N. Elgin, “Stochastic perturbations of optical solitons,” Opt. Lett. 18, 10 (1993).
    [CrossRef] [PubMed]
  3. H. A. Haus, “Quantum noise in a solitonlike repeater system,” J. Opt. Soc. Am. B 8, 1122 (1991).
    [CrossRef]
  4. A. Mecozzi, “Long distance transmission at zero dispersion: combined effect of Kerr nonlinearity and the noise of the in-line amplifiers,” J. Opt. Soc. Am. B 11, 462 (1994).
    [CrossRef]
  5. F. Kh. Abdullaev, J. G. Caputo, and N. Flytzanis, “Envelope soliton propagation in media with temporally varying dispersion,” Phys. Rev. E 50, 1552 (1994).
    [CrossRef]
  6. S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Laser Pulses (American Institute of Physics, New York, 1992).
  7. J. P. Gordon, “Dispersive-perturbation of solitons of the nonlinear Schrödinger equation,” J. Opt. Soc. Am. B 9, 9 (1992).
    [CrossRef]
  8. R. G. Bauer and L. A. Mel’nikov, “Multisoliton fission and quasi-periodicity in a fiber with a periodically modulated core diameter,” Optics Commun. 115, 190 (1995).
    [CrossRef]
  9. B. A. Malomed, “Resonant amplification of a chirped soliton in a long optical fiber with periodic amplification,” J. Opt. Soc. Am. B 13, 677 (1996).
    [CrossRef]
  10. F. Kh. Abdullaev, A. A. Abdumalikov, and B. B. Baizakov, “Stochastic instability of chirped optical solitons in media with periodic amplification,” Quantum Electron. 27, 171 (1997).
    [CrossRef]
  11. F. Kh. Abdullaev, A. A. Abdumalikov, and B. B. Baizakov, “Propagation of chirped optical solitons in fibers with randomly varying parameters,” Opt. Commun. 138, 49 (1997).
    [CrossRef]
  12. G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1990).
  13. F. Kh. Abdullaev, S. A. Darmanyan, and P. K. Khabibullaev, Optical Solitons (Springer-Verlag, Heidelberg, 1993).
  14. F. Kh. Abdullaev, “Propagation of a soliton in a fiber with fluctuating parameters,” Sov. Tech. Phys. Lett. 9, 305 (1983).
  15. V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevsky, Theory of Solitons. Inverse Scattering Method (Consultants Bureau, New York, 1984).
  16. Yu. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763 (1989).
    [CrossRef]
  17. V. V. Konotop and L. Vasquez, Nonlinear Random Waves (World Scientific, Singapore, 1994).
  18. S. A. Gredeskul and Yu. S. Kivshar, “Propagation and scattering of nonlinear waves in disordered systems,” Phys. Rep. 216, 1 (1992).
    [CrossRef]
  19. Yu. S. Kivshar, V. V. Konotop, and Yu. A. Sinitsyn, “Emission of solitons in a fluctuating medium,” Radiofizika 30, 374 (1987).
  20. V. I. Karpman and V. V. Solov’ev, “A perturbation theory for soliton systems,” Physica D 3, 142 (1981).
    [CrossRef]
  21. B. A. Malomed, “Bound states in a gas of solitons supported by a randomly fluctuating force,” Europhys. Lett. 30, 507 (1995).
    [CrossRef]
  22. H. Risken, The Fokker-Planck Equation (Springer-Verlag, Heidelberg, 1984).
  23. R. Grimshaw, J. He, and B. Malomed, “Decay of a fundamental soliton in a periodically modulated nonlinear waveguide,” Phys. Scr. 53, 385 (1996).
    [CrossRef]
  24. V. A. Hopkins, J. Keat, G. D. Meegan, T. Zhang, and J. D. Maynard, “Observation of the predicted behavior of nonlinear pulse propagation in disordered media,” Phys. Rev. Lett. 76, 1102 (1996).
    [CrossRef] [PubMed]
  25. S. Molchanov, Lectures on Random Media, Vol. 1581 of Springer Lecture Notes in Mathematics (Springer-Verlag, Berlin, 1994), Lecture 7.

1997 (2)

F. Kh. Abdullaev, A. A. Abdumalikov, and B. B. Baizakov, “Stochastic instability of chirped optical solitons in media with periodic amplification,” Quantum Electron. 27, 171 (1997).
[CrossRef]

F. Kh. Abdullaev, A. A. Abdumalikov, and B. B. Baizakov, “Propagation of chirped optical solitons in fibers with randomly varying parameters,” Opt. Commun. 138, 49 (1997).
[CrossRef]

1996 (3)

B. A. Malomed, “Resonant amplification of a chirped soliton in a long optical fiber with periodic amplification,” J. Opt. Soc. Am. B 13, 677 (1996).
[CrossRef]

R. Grimshaw, J. He, and B. Malomed, “Decay of a fundamental soliton in a periodically modulated nonlinear waveguide,” Phys. Scr. 53, 385 (1996).
[CrossRef]

V. A. Hopkins, J. Keat, G. D. Meegan, T. Zhang, and J. D. Maynard, “Observation of the predicted behavior of nonlinear pulse propagation in disordered media,” Phys. Rev. Lett. 76, 1102 (1996).
[CrossRef] [PubMed]

1995 (2)

B. A. Malomed, “Bound states in a gas of solitons supported by a randomly fluctuating force,” Europhys. Lett. 30, 507 (1995).
[CrossRef]

R. G. Bauer and L. A. Mel’nikov, “Multisoliton fission and quasi-periodicity in a fiber with a periodically modulated core diameter,” Optics Commun. 115, 190 (1995).
[CrossRef]

1994 (2)

A. Mecozzi, “Long distance transmission at zero dispersion: combined effect of Kerr nonlinearity and the noise of the in-line amplifiers,” J. Opt. Soc. Am. B 11, 462 (1994).
[CrossRef]

F. Kh. Abdullaev, J. G. Caputo, and N. Flytzanis, “Envelope soliton propagation in media with temporally varying dispersion,” Phys. Rev. E 50, 1552 (1994).
[CrossRef]

1993 (1)

1992 (2)

J. P. Gordon, “Dispersive-perturbation of solitons of the nonlinear Schrödinger equation,” J. Opt. Soc. Am. B 9, 9 (1992).
[CrossRef]

S. A. Gredeskul and Yu. S. Kivshar, “Propagation and scattering of nonlinear waves in disordered systems,” Phys. Rep. 216, 1 (1992).
[CrossRef]

1991 (1)

1989 (1)

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

1987 (1)

Yu. S. Kivshar, V. V. Konotop, and Yu. A. Sinitsyn, “Emission of solitons in a fluctuating medium,” Radiofizika 30, 374 (1987).

1983 (1)

F. Kh. Abdullaev, “Propagation of a soliton in a fiber with fluctuating parameters,” Sov. Tech. Phys. Lett. 9, 305 (1983).

1981 (1)

V. I. Karpman and V. V. Solov’ev, “A perturbation theory for soliton systems,” Physica D 3, 142 (1981).
[CrossRef]

Abdullaev, F. Kh.

F. Kh. Abdullaev, A. A. Abdumalikov, and B. B. Baizakov, “Stochastic instability of chirped optical solitons in media with periodic amplification,” Quantum Electron. 27, 171 (1997).
[CrossRef]

F. Kh. Abdullaev, A. A. Abdumalikov, and B. B. Baizakov, “Propagation of chirped optical solitons in fibers with randomly varying parameters,” Opt. Commun. 138, 49 (1997).
[CrossRef]

F. Kh. Abdullaev, J. G. Caputo, and N. Flytzanis, “Envelope soliton propagation in media with temporally varying dispersion,” Phys. Rev. E 50, 1552 (1994).
[CrossRef]

F. Kh. Abdullaev, “Propagation of a soliton in a fiber with fluctuating parameters,” Sov. Tech. Phys. Lett. 9, 305 (1983).

Abdumalikov, A. A.

F. Kh. Abdullaev, A. A. Abdumalikov, and B. B. Baizakov, “Propagation of chirped optical solitons in fibers with randomly varying parameters,” Opt. Commun. 138, 49 (1997).
[CrossRef]

F. Kh. Abdullaev, A. A. Abdumalikov, and B. B. Baizakov, “Stochastic instability of chirped optical solitons in media with periodic amplification,” Quantum Electron. 27, 171 (1997).
[CrossRef]

Baizakov, B. B.

F. Kh. Abdullaev, A. A. Abdumalikov, and B. B. Baizakov, “Stochastic instability of chirped optical solitons in media with periodic amplification,” Quantum Electron. 27, 171 (1997).
[CrossRef]

F. Kh. Abdullaev, A. A. Abdumalikov, and B. B. Baizakov, “Propagation of chirped optical solitons in fibers with randomly varying parameters,” Opt. Commun. 138, 49 (1997).
[CrossRef]

Bauer, R. G.

R. G. Bauer and L. A. Mel’nikov, “Multisoliton fission and quasi-periodicity in a fiber with a periodically modulated core diameter,” Optics Commun. 115, 190 (1995).
[CrossRef]

Caputo, J. G.

F. Kh. Abdullaev, J. G. Caputo, and N. Flytzanis, “Envelope soliton propagation in media with temporally varying dispersion,” Phys. Rev. E 50, 1552 (1994).
[CrossRef]

Elgin, J. N.

Flytzanis, N.

F. Kh. Abdullaev, J. G. Caputo, and N. Flytzanis, “Envelope soliton propagation in media with temporally varying dispersion,” Phys. Rev. E 50, 1552 (1994).
[CrossRef]

Gordon, J. P.

J. P. Gordon, “Dispersive-perturbation of solitons of the nonlinear Schrödinger equation,” J. Opt. Soc. Am. B 9, 9 (1992).
[CrossRef]

Gredeskul, S. A.

S. A. Gredeskul and Yu. S. Kivshar, “Propagation and scattering of nonlinear waves in disordered systems,” Phys. Rep. 216, 1 (1992).
[CrossRef]

Grimshaw, R.

R. Grimshaw, J. He, and B. Malomed, “Decay of a fundamental soliton in a periodically modulated nonlinear waveguide,” Phys. Scr. 53, 385 (1996).
[CrossRef]

Haus, H. A.

He, J.

R. Grimshaw, J. He, and B. Malomed, “Decay of a fundamental soliton in a periodically modulated nonlinear waveguide,” Phys. Scr. 53, 385 (1996).
[CrossRef]

Hopkins, V. A.

V. A. Hopkins, J. Keat, G. D. Meegan, T. Zhang, and J. D. Maynard, “Observation of the predicted behavior of nonlinear pulse propagation in disordered media,” Phys. Rev. Lett. 76, 1102 (1996).
[CrossRef] [PubMed]

Karpman, V. I.

V. I. Karpman and V. V. Solov’ev, “A perturbation theory for soliton systems,” Physica D 3, 142 (1981).
[CrossRef]

Keat, J.

V. A. Hopkins, J. Keat, G. D. Meegan, T. Zhang, and J. D. Maynard, “Observation of the predicted behavior of nonlinear pulse propagation in disordered media,” Phys. Rev. Lett. 76, 1102 (1996).
[CrossRef] [PubMed]

Kivshar, Yu. S.

S. A. Gredeskul and Yu. S. Kivshar, “Propagation and scattering of nonlinear waves in disordered systems,” Phys. Rep. 216, 1 (1992).
[CrossRef]

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

Yu. S. Kivshar, V. V. Konotop, and Yu. A. Sinitsyn, “Emission of solitons in a fluctuating medium,” Radiofizika 30, 374 (1987).

Konotop, V. V.

Yu. S. Kivshar, V. V. Konotop, and Yu. A. Sinitsyn, “Emission of solitons in a fluctuating medium,” Radiofizika 30, 374 (1987).

Malomed, B.

R. Grimshaw, J. He, and B. Malomed, “Decay of a fundamental soliton in a periodically modulated nonlinear waveguide,” Phys. Scr. 53, 385 (1996).
[CrossRef]

Malomed, B. A.

B. A. Malomed, “Resonant amplification of a chirped soliton in a long optical fiber with periodic amplification,” J. Opt. Soc. Am. B 13, 677 (1996).
[CrossRef]

B. A. Malomed, “Bound states in a gas of solitons supported by a randomly fluctuating force,” Europhys. Lett. 30, 507 (1995).
[CrossRef]

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

Maynard, J. D.

V. A. Hopkins, J. Keat, G. D. Meegan, T. Zhang, and J. D. Maynard, “Observation of the predicted behavior of nonlinear pulse propagation in disordered media,” Phys. Rev. Lett. 76, 1102 (1996).
[CrossRef] [PubMed]

Mecozzi, A.

Meegan, G. D.

V. A. Hopkins, J. Keat, G. D. Meegan, T. Zhang, and J. D. Maynard, “Observation of the predicted behavior of nonlinear pulse propagation in disordered media,” Phys. Rev. Lett. 76, 1102 (1996).
[CrossRef] [PubMed]

Mel’nikov, L. A.

R. G. Bauer and L. A. Mel’nikov, “Multisoliton fission and quasi-periodicity in a fiber with a periodically modulated core diameter,” Optics Commun. 115, 190 (1995).
[CrossRef]

Sinitsyn, Yu. A.

Yu. S. Kivshar, V. V. Konotop, and Yu. A. Sinitsyn, “Emission of solitons in a fluctuating medium,” Radiofizika 30, 374 (1987).

Solov’ev, V. V.

V. I. Karpman and V. V. Solov’ev, “A perturbation theory for soliton systems,” Physica D 3, 142 (1981).
[CrossRef]

Zhang, T.

V. A. Hopkins, J. Keat, G. D. Meegan, T. Zhang, and J. D. Maynard, “Observation of the predicted behavior of nonlinear pulse propagation in disordered media,” Phys. Rev. Lett. 76, 1102 (1996).
[CrossRef] [PubMed]

Europhys. Lett. (1)

B. A. Malomed, “Bound states in a gas of solitons supported by a randomly fluctuating force,” Europhys. Lett. 30, 507 (1995).
[CrossRef]

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

Opt. Commun. (1)

F. Kh. Abdullaev, A. A. Abdumalikov, and B. B. Baizakov, “Propagation of chirped optical solitons in fibers with randomly varying parameters,” Opt. Commun. 138, 49 (1997).
[CrossRef]

Opt. Lett. (1)

Optics Commun. (1)

R. G. Bauer and L. A. Mel’nikov, “Multisoliton fission and quasi-periodicity in a fiber with a periodically modulated core diameter,” Optics Commun. 115, 190 (1995).
[CrossRef]

Phys. Rep. (1)

S. A. Gredeskul and Yu. S. Kivshar, “Propagation and scattering of nonlinear waves in disordered systems,” Phys. Rep. 216, 1 (1992).
[CrossRef]

Phys. Rev. E (1)

F. Kh. Abdullaev, J. G. Caputo, and N. Flytzanis, “Envelope soliton propagation in media with temporally varying dispersion,” Phys. Rev. E 50, 1552 (1994).
[CrossRef]

Phys. Rev. Lett. (1)

V. A. Hopkins, J. Keat, G. D. Meegan, T. Zhang, and J. D. Maynard, “Observation of the predicted behavior of nonlinear pulse propagation in disordered media,” Phys. Rev. Lett. 76, 1102 (1996).
[CrossRef] [PubMed]

Phys. Scr. (1)

R. Grimshaw, J. He, and B. Malomed, “Decay of a fundamental soliton in a periodically modulated nonlinear waveguide,” Phys. Scr. 53, 385 (1996).
[CrossRef]

Physica D (1)

V. I. Karpman and V. V. Solov’ev, “A perturbation theory for soliton systems,” Physica D 3, 142 (1981).
[CrossRef]

Quantum Electron. (1)

F. Kh. Abdullaev, A. A. Abdumalikov, and B. B. Baizakov, “Stochastic instability of chirped optical solitons in media with periodic amplification,” Quantum Electron. 27, 171 (1997).
[CrossRef]

Radiofizika (1)

Yu. S. Kivshar, V. V. Konotop, and Yu. A. Sinitsyn, “Emission of solitons in a fluctuating medium,” Radiofizika 30, 374 (1987).

Rev. Mod. Phys. (1)

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

Sov. Tech. Phys. Lett. (1)

F. Kh. Abdullaev, “Propagation of a soliton in a fiber with fluctuating parameters,” Sov. Tech. Phys. Lett. 9, 305 (1983).

Other (8)

V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevsky, Theory of Solitons. Inverse Scattering Method (Consultants Bureau, New York, 1984).

V. V. Konotop and L. Vasquez, Nonlinear Random Waves (World Scientific, Singapore, 1994).

G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1990).

F. Kh. Abdullaev, S. A. Darmanyan, and P. K. Khabibullaev, Optical Solitons (Springer-Verlag, Heidelberg, 1993).

S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Laser Pulses (American Institute of Physics, New York, 1992).

A. Hasegawa and Yu. Kodama, Solitons in Optical Communications (Clarendon, Oxford, 1995).

H. Risken, The Fokker-Planck Equation (Springer-Verlag, Heidelberg, 1984).

S. Molchanov, Lectures on Random Media, Vol. 1581 of Springer Lecture Notes in Mathematics (Springer-Verlag, Berlin, 1994), Lecture 7.

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Figures (9)

Fig. 1
Fig. 1

One realization of the propagation of a soliton in a stochastic medium with white-noise perturbation and σ2=0.1 [Eq. (1)]. The initial condition [Eq. (4)] is 2η0=1.0.

Fig. 2
Fig. 2

Mean value max|u|2 as a function of position x with white noise in Eqs. (2). The average is taken over 150 realizations (solid curves). The numerical simulations are obtained by use of the split-step Fourier method. The analytical solutions correspond to Eq. (15) (dashed curves). σ2=0.01 (top curves), 0.1 (middle curves), and 0.2 (bottom curves). The initial condition is 2η0=1.0.

Fig. 3
Fig. 3

Mean value max|u|2 as a function of position x with white noise. Solid curve, numerical simulation of Eq. (1); dashed curve, analytical solutions from Eq. (15). The initial conditions are 2η0=2.0 and σ2=0.1.

Fig. 4
Fig. 4

Mean value max|u|2 as a function of position x with colored noise in Eq. (1), where B(x-x)=σ12/(2γ)exp(-γ|x-x|) for three values of γ and σ2=σ12/γ2=0.1. Note that γ is the inverse of the correlation length. Dashed curve, white-noise case with σ2=0.1.

Fig. 5
Fig. 5

Decay of the soliton for white noise with σ2=0.125. The initial condition is 2η0=1.3.

Fig. 6
Fig. 6

Stabilization of a soliton owing to colored noise with σ2=σ12/γ2=0.125 and γ=1.0. The initial condition is 2η0=1.3.

Fig. 7
Fig. 7

Soliton amplitude difference calculated from Eq. (1) with colored-noise and the two-soliton initial condition. σ12=1.0, γ=1.0, a1=1.0, and a2=0.8.

Fig. 8
Fig. 8

After one interaction the two-soliton state decays under white noise with σ2=0.01. In the initial state t0=5 and a1=a2=1.0.

Fig. 9
Fig. 9

Two-soliton interaction and the appearance of a bound state owing to colored noise with σ12=0.05 and γ=0.87. In the initial state t0=5 and a1=a2=1.0.

Equations (56)

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iux+utt+2|u|2u=-V(x)|u|2u,
V(x)=0,V(x)V(x)=B(x-x; lc),
iux+[1+(x)]utt+2|u|2u=0.
u(x, t)=2iη sech(z)exp[-iξz/η+iϕ)],
η=η0,ξ=ξ0,ζ=-4ξ0x,
dϕdx=4(ξ2-η2)-23 η2V(x).
(Δϕ)2=(ϕ-ϕ0)2=49 σ2η4L,
B(x-x; lc)=σ22lc exp-|x-x|lc.
(Δϕ)2=49 η4σ2{L+lc[exp(-L/lc)-1]}.
N=-|u|2dt=4η+1π - ln[1-|b(x, λ)|2]dλ,
dbdx=-4iλ2b(x, λ)-iV(x)exp(iϕ-2iΔζ)2η(Δ2+η2) A(λ, ξ, x),
A=-dz[Δ-iη tanh(z)]2R(u)exp(-iθ)-η2cosh2 z R*(u)exp(iθ)exp-i Δη z.
P(λ)=2 Reb(λ) db*dx,2λ=k.
A(λ, ξ)=-2πiη(Δ2+η2)2coshπΔ2η.
dηdx=-12π 0P(λ)dλ=-σ2η5 6415.
η(x)=η0(1+κx)-1/4,κ=4σ2μη04,
μ=64/15.
Q=i2 -dt[u*ut-utu*]=const.
Q=8(ξη)+2π -2λP(Δ)dλ,Δ=λ-ξ.
P(λ)=π2σ2[Δ2+η2]2[1+16lc2(Δ2+η2)2]cosh2πλ2η.
dηdx-πσ216lc2 η.
η=η0 exp-πσ216lc2 x.
u=us(t+t0)+exp(iψ)us(t-t0).
dpdx=64η3 exp(-2ηr)sin(2ξr+ψ),
dψdx=8η[1+2V(x)]p,
drdx=-4q,
dqdx=64η3 exp(-2ηr)cos(2ξr+ψ).
η=η1+η2=const.,ξ=ξ1+ξ2=const.
rxx=-256η3 exp(-2ηr)cos(2ξr+ψ).
T=4 exp-t0k+A-1A+121/2,k=A+14A.
p=p0+p1+,ψ=ψ0+ψ1+,
q=q0+q1+,r=r0+r1+ .
dp1dx=64η3 exp(-2ηr0)cos(ψ0)ψ1-128η4×exp(-2ηr0)sin(ψ0)r1,
dψ1dx=8ηp1+16ηp0V(x),
dq1dx=-64η3 exp(-2ηr0)×[sin(ψ0)r1+2η cos(ψ0)r1],
dr1dx=-4q1.
d2p1dx2=-512η4 exp(-2ηr0)[2p0V(x)+p1],
d2r1dx2=512η4 exp(-2ηr0)r1.
p124ω02p020x0xdxdxV(x)V(x)×sin[ω0(x-x)]sin[ω0(x-x)].
p122ω02p02σ2x-sin(2ω0x)2ω0.
p12=4ω02p02σ2γγ2+ω02 γx2-γ4ω0sin(2ω0x)+12 sin2(ω0x)-1γ2+ω02×[exp(-γx)ω0[γ sin(ω0x)+ω0 cos (ω0x)]+γ2 sin2(ω0x)-ω02 cos2(ω0x)],
p122ω02p02σ2x1+ω02lc2 1-sin(2ω0x)2ω0x.
U=128η2 exp(-2ηr)cos ψ,
U=128η2 exp[-(2ηr)]exp(-Ψ2/2)=128η2 exp(-σΨ2x),
ux=iutt,
ux=i2|u|2u+i|u|2uV(x).
u(x+dx, t)=F-1{F[u(x, t)]exp(-iω2dx)},
u(x+dx, t)=u(x, t)×expi2|u|2+|u|2 1dx V(x)dx,
Vn=0,
VnVn=σ122γ exp(-γ|xn-xn|).
dVdx+γV=Γ(x),
Vn+1=Vn-γVndx+dxσ12/2wn,
u(xn+1, t)=u(xn, t)×expi2|u|2+|u|2 Vn+Vn+12dx.
u(x, t)=a1 sech(t+t0)+a2 exp(iψ0)sech(t-t0).
u(x, t)=A(x)sech[t/a(x)]exp[b(x)t2].
t0=πηL,

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