Abstract

Oscillating tails of a dispersion-managed optical fiber system are studied for a strong dispersion map in the framework of a path-averaged Gabitov–Turitsyn equation. The small parameter of the analytical theory is the inverse time. An exponential decay in time of a soliton tail envelope is consistent with nonlocal nonlinearity of the Gabitov–Turitsyn equation, and the fast oscillations are described by a quadratic law. The preexponential modification factor is the linear function of time for zero average dispersion and a cubic function for nonzero average dispersion.

© 2004 Optical Society of America

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  1. C. Lin, H. Kogelnik, and L. G. Cohen, “Optical-pulse equalization of low-dispersion transmission in single-mode fibers in the 1.3–1.7-μm spectral region,” Opt. Lett. 5, 476–478 (1980).
    [CrossRef] [PubMed]
  2. C. Kurtzke, “Suppression of fiber nonlinearities by appropriate dispersion management,” IEEE Photonics Technol. Lett. 5, 1250–1253 (1993).
    [CrossRef]
  3. A. R. Chraplyvy, A. H. Gnauck, R. W. Tkach, and R. M. Derosier, “8×10 Gb/s transmission through 280 km of dispersion-managed fiber,” IEEE Photonics Technol. Lett. 5, 1233–1235 (1993).
    [CrossRef]
  4. N. J. Smith, F. M. Knox, N. J. Doran, K. J. Blow, and I. Bennion, “Enhanced power solitons in optical fiber transmission line,” Electron. Lett. 32, 54–55 (1996).
    [CrossRef]
  5. I. Gabitov and S. K. Turitsyn, “Averaged pulse dynamics in a cascaded transmission system with passive dispersion compensation,” Opt. Lett. 21, 327–329 (1996).
    [CrossRef]
  6. I. Gabitov and S. K. Turitsyn, “Breathing solitons in optical fiber links,” JETP Lett. 63, 861–866 (1996).
    [CrossRef]
  7. S. Kumar and A. Hasegawa, “Quasi-soliton propagation in dispersion-managed optical fibers,” Opt. Lett. 22, 372–374 (1997).
    [CrossRef] [PubMed]
  8. T. Lakoba and D. J. Kaup, “Shape of the stationary pulse in the strong dispersion management regime,” Electron. Lett. 34, 1124–1125 (1998).
    [CrossRef]
  9. P. V. Mamyshev and N. A. Mamysheva, “Pulse-overlapped dispersion-managed data transmission and intrachannel four-wave mixing,” Opt. Lett. 24, 1454–1456 (1999).
    [CrossRef]
  10. L. F. Mollenauer, P. V. Mamyshev, J. Gripp, M. J. Neubelt, N. Mamysheva, L. Grüner-Nielsen, and T. Veng, “Demonstration of massive wavelength-division multiplexing over transoceanic distances by use of dispersion-managed solitons,” Opt. Lett. 25, 704–706 (2000).
    [CrossRef]
  11. S. K. Turitsyn, T. Schäfer, N. J. Doran, K. H. Spatschek, and V. K. Mezentsev, “Pathaveraged chirped optical soliton in dispersion-managed fiber communication lines,” Opt. Commun. 163, 122–158 (1999).
    [CrossRef]
  12. S. B. Medvedev, O. V. Shtyrina, S. L. Musher, and M. P. Fedoruk, “Path-averaged optical soliton in double-periodic dispersion-managed systems,” Phys. Rev. E 66, 066607 (2002).
    [CrossRef]
  13. A. Gajadharsingh and P.-A. Bélanger, “Dispersion-managed solitons as the interference of chirped complex conjugate pulses,” in Nonlinear Guided Waves and their Applications, 2004 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 2004), paper MC7.
  14. B. Biotteau, J. P. Hamaide, F. Pitel, O. Audoin, P. Nouchi, and P. Sansonetti, “Enhancement of soliton system performance by use of new large effective area fibres,” Electron. Lett. 31, 2026–2027 (1995).
  15. C. Weinstein, “Fiber design improves long-haul performance,” Laser Focus World 33, 215 (1997).
  16. C. Paré, A. Villeneuve, P.-A. Bélanger, and N. J. Doran, “Compensating for dispersion and the nonlinear Kerr effect without phase conjugation,” Opt. Lett. 21, 459–461 (1996).
    [CrossRef] [PubMed]
  17. I. R. Gabitov and P. M. Lushnikov, “Nonlinearity management in a dispersion-managed system,” Opt. Lett. 27, 113–115 (2002).
    [CrossRef]
  18. L. F. Mollenauer, A. Grant, X. Liu, X. Wei, C. Xie, and I. Kang, “Experimental test of dense wavelength-division multiplexing using novel, periodic-group-delay-complemented dispersion compensation and dispersion-managed solitons,” Opt. Lett. 28, 2043–2045 (2003).
    [CrossRef] [PubMed]
  19. X. Wei, X. Liu, C. Xie, and L. F. Mollenauer, “Reduction of collision-induced timing jitter in dense wavelength-division multiplexing by the use of periodic-group-delay dispersion compensators,” Opt. Lett. 28, 983–985 (2003).
    [CrossRef] [PubMed]
  20. M. J. Ablowitz and G. Biondini, “Multiscale pulse dynamics in communication systems with strong dispersion management,” Opt. Lett. 23, 1668–1670 (1998).
    [CrossRef]
  21. P. M. Lushnikov, “Dispersion-managed soliton in optical fibers with zero average dispersion,” Opt. Lett. 25, 1144–1146 (2000).
    [CrossRef]
  22. P. M. Lushnikov, “On the boundary of the dispersion-managed soliton existence,” JETP Lett. 72, 111–114 (2000).
    [CrossRef]
  23. D. E. Pelinovsky, “Instabilities of dispersion-managed solitons in the normal dispersion regime,” Phys. Rev. E 62, 4283–4293 (2000).
    [CrossRef]
  24. V. Zharnitsky, E. Grenier, C. K. R. T. Jones, and S. K. Turitsyn, “Stabilizing effects of dispersion management,” Physica D 152, 794–817 (2001).
    [CrossRef]
  25. D. E. Pelinovsky and V. Zharnitsky, “Averaging of dispersion-managed solitons: existence and stability,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 63, 745–776 (2003).
    [CrossRef]
  26. M. Kunze, “The singular perturbation limit of a variational problem from nonlinear fiber optics,” Physica D 180, 108–114 (2003).
    [CrossRef]
  27. P. M. Lushnikov, “Dispersion-managed soliton in a strong dispersion map limit,” Opt. Lett. 26, 1535–1537 (2001).
    [CrossRef]
  28. P. M. Lushnikov, “Fully parallel algorithm for simulating wavelength-division-multiplexed optical fiber systems,” Opt. Lett. 27, 939–941 (2002).
    [CrossRef]

2003

2002

2001

V. Zharnitsky, E. Grenier, C. K. R. T. Jones, and S. K. Turitsyn, “Stabilizing effects of dispersion management,” Physica D 152, 794–817 (2001).
[CrossRef]

P. M. Lushnikov, “Dispersion-managed soliton in a strong dispersion map limit,” Opt. Lett. 26, 1535–1537 (2001).
[CrossRef]

2000

1999

S. K. Turitsyn, T. Schäfer, N. J. Doran, K. H. Spatschek, and V. K. Mezentsev, “Pathaveraged chirped optical soliton in dispersion-managed fiber communication lines,” Opt. Commun. 163, 122–158 (1999).
[CrossRef]

P. V. Mamyshev and N. A. Mamysheva, “Pulse-overlapped dispersion-managed data transmission and intrachannel four-wave mixing,” Opt. Lett. 24, 1454–1456 (1999).
[CrossRef]

1998

M. J. Ablowitz and G. Biondini, “Multiscale pulse dynamics in communication systems with strong dispersion management,” Opt. Lett. 23, 1668–1670 (1998).
[CrossRef]

T. Lakoba and D. J. Kaup, “Shape of the stationary pulse in the strong dispersion management regime,” Electron. Lett. 34, 1124–1125 (1998).
[CrossRef]

1997

1996

N. J. Smith, F. M. Knox, N. J. Doran, K. J. Blow, and I. Bennion, “Enhanced power solitons in optical fiber transmission line,” Electron. Lett. 32, 54–55 (1996).
[CrossRef]

I. Gabitov and S. K. Turitsyn, “Breathing solitons in optical fiber links,” JETP Lett. 63, 861–866 (1996).
[CrossRef]

I. Gabitov and S. K. Turitsyn, “Averaged pulse dynamics in a cascaded transmission system with passive dispersion compensation,” Opt. Lett. 21, 327–329 (1996).
[CrossRef]

C. Paré, A. Villeneuve, P.-A. Bélanger, and N. J. Doran, “Compensating for dispersion and the nonlinear Kerr effect without phase conjugation,” Opt. Lett. 21, 459–461 (1996).
[CrossRef] [PubMed]

1993

C. Kurtzke, “Suppression of fiber nonlinearities by appropriate dispersion management,” IEEE Photonics Technol. Lett. 5, 1250–1253 (1993).
[CrossRef]

A. R. Chraplyvy, A. H. Gnauck, R. W. Tkach, and R. M. Derosier, “8×10 Gb/s transmission through 280 km of dispersion-managed fiber,” IEEE Photonics Technol. Lett. 5, 1233–1235 (1993).
[CrossRef]

1980

Ablowitz, M. J.

Bélanger, P.-A.

Bennion, I.

N. J. Smith, F. M. Knox, N. J. Doran, K. J. Blow, and I. Bennion, “Enhanced power solitons in optical fiber transmission line,” Electron. Lett. 32, 54–55 (1996).
[CrossRef]

Biondini, G.

Blow, K. J.

N. J. Smith, F. M. Knox, N. J. Doran, K. J. Blow, and I. Bennion, “Enhanced power solitons in optical fiber transmission line,” Electron. Lett. 32, 54–55 (1996).
[CrossRef]

Chraplyvy, A. R.

A. R. Chraplyvy, A. H. Gnauck, R. W. Tkach, and R. M. Derosier, “8×10 Gb/s transmission through 280 km of dispersion-managed fiber,” IEEE Photonics Technol. Lett. 5, 1233–1235 (1993).
[CrossRef]

Cohen, L. G.

Derosier, R. M.

A. R. Chraplyvy, A. H. Gnauck, R. W. Tkach, and R. M. Derosier, “8×10 Gb/s transmission through 280 km of dispersion-managed fiber,” IEEE Photonics Technol. Lett. 5, 1233–1235 (1993).
[CrossRef]

Doran, N. J.

S. K. Turitsyn, T. Schäfer, N. J. Doran, K. H. Spatschek, and V. K. Mezentsev, “Pathaveraged chirped optical soliton in dispersion-managed fiber communication lines,” Opt. Commun. 163, 122–158 (1999).
[CrossRef]

N. J. Smith, F. M. Knox, N. J. Doran, K. J. Blow, and I. Bennion, “Enhanced power solitons in optical fiber transmission line,” Electron. Lett. 32, 54–55 (1996).
[CrossRef]

C. Paré, A. Villeneuve, P.-A. Bélanger, and N. J. Doran, “Compensating for dispersion and the nonlinear Kerr effect without phase conjugation,” Opt. Lett. 21, 459–461 (1996).
[CrossRef] [PubMed]

Fedoruk, M. P.

S. B. Medvedev, O. V. Shtyrina, S. L. Musher, and M. P. Fedoruk, “Path-averaged optical soliton in double-periodic dispersion-managed systems,” Phys. Rev. E 66, 066607 (2002).
[CrossRef]

Gabitov, I.

Gabitov, I. R.

Gnauck, A. H.

A. R. Chraplyvy, A. H. Gnauck, R. W. Tkach, and R. M. Derosier, “8×10 Gb/s transmission through 280 km of dispersion-managed fiber,” IEEE Photonics Technol. Lett. 5, 1233–1235 (1993).
[CrossRef]

Grant, A.

Grenier, E.

V. Zharnitsky, E. Grenier, C. K. R. T. Jones, and S. K. Turitsyn, “Stabilizing effects of dispersion management,” Physica D 152, 794–817 (2001).
[CrossRef]

Gripp, J.

Grüner-Nielsen, L.

Hasegawa, A.

Jones, C. K. R. T.

V. Zharnitsky, E. Grenier, C. K. R. T. Jones, and S. K. Turitsyn, “Stabilizing effects of dispersion management,” Physica D 152, 794–817 (2001).
[CrossRef]

Kang, I.

Kaup, D. J.

T. Lakoba and D. J. Kaup, “Shape of the stationary pulse in the strong dispersion management regime,” Electron. Lett. 34, 1124–1125 (1998).
[CrossRef]

Knox, F. M.

N. J. Smith, F. M. Knox, N. J. Doran, K. J. Blow, and I. Bennion, “Enhanced power solitons in optical fiber transmission line,” Electron. Lett. 32, 54–55 (1996).
[CrossRef]

Kogelnik, H.

Kumar, S.

Kunze, M.

M. Kunze, “The singular perturbation limit of a variational problem from nonlinear fiber optics,” Physica D 180, 108–114 (2003).
[CrossRef]

Kurtzke, C.

C. Kurtzke, “Suppression of fiber nonlinearities by appropriate dispersion management,” IEEE Photonics Technol. Lett. 5, 1250–1253 (1993).
[CrossRef]

Lakoba, T.

T. Lakoba and D. J. Kaup, “Shape of the stationary pulse in the strong dispersion management regime,” Electron. Lett. 34, 1124–1125 (1998).
[CrossRef]

Lin, C.

Liu, X.

Lushnikov, P. M.

Mamyshev, P. V.

Mamysheva, N.

Mamysheva, N. A.

Medvedev, S. B.

S. B. Medvedev, O. V. Shtyrina, S. L. Musher, and M. P. Fedoruk, “Path-averaged optical soliton in double-periodic dispersion-managed systems,” Phys. Rev. E 66, 066607 (2002).
[CrossRef]

Mezentsev, V. K.

S. K. Turitsyn, T. Schäfer, N. J. Doran, K. H. Spatschek, and V. K. Mezentsev, “Pathaveraged chirped optical soliton in dispersion-managed fiber communication lines,” Opt. Commun. 163, 122–158 (1999).
[CrossRef]

Mollenauer, L. F.

Musher, S. L.

S. B. Medvedev, O. V. Shtyrina, S. L. Musher, and M. P. Fedoruk, “Path-averaged optical soliton in double-periodic dispersion-managed systems,” Phys. Rev. E 66, 066607 (2002).
[CrossRef]

Neubelt, M. J.

Paré, C.

Pelinovsky, D. E.

D. E. Pelinovsky and V. Zharnitsky, “Averaging of dispersion-managed solitons: existence and stability,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 63, 745–776 (2003).
[CrossRef]

D. E. Pelinovsky, “Instabilities of dispersion-managed solitons in the normal dispersion regime,” Phys. Rev. E 62, 4283–4293 (2000).
[CrossRef]

Schäfer, T.

S. K. Turitsyn, T. Schäfer, N. J. Doran, K. H. Spatschek, and V. K. Mezentsev, “Pathaveraged chirped optical soliton in dispersion-managed fiber communication lines,” Opt. Commun. 163, 122–158 (1999).
[CrossRef]

Shtyrina, O. V.

S. B. Medvedev, O. V. Shtyrina, S. L. Musher, and M. P. Fedoruk, “Path-averaged optical soliton in double-periodic dispersion-managed systems,” Phys. Rev. E 66, 066607 (2002).
[CrossRef]

Smith, N. J.

N. J. Smith, F. M. Knox, N. J. Doran, K. J. Blow, and I. Bennion, “Enhanced power solitons in optical fiber transmission line,” Electron. Lett. 32, 54–55 (1996).
[CrossRef]

Spatschek, K. H.

S. K. Turitsyn, T. Schäfer, N. J. Doran, K. H. Spatschek, and V. K. Mezentsev, “Pathaveraged chirped optical soliton in dispersion-managed fiber communication lines,” Opt. Commun. 163, 122–158 (1999).
[CrossRef]

Tkach, R. W.

A. R. Chraplyvy, A. H. Gnauck, R. W. Tkach, and R. M. Derosier, “8×10 Gb/s transmission through 280 km of dispersion-managed fiber,” IEEE Photonics Technol. Lett. 5, 1233–1235 (1993).
[CrossRef]

Turitsyn, S. K.

V. Zharnitsky, E. Grenier, C. K. R. T. Jones, and S. K. Turitsyn, “Stabilizing effects of dispersion management,” Physica D 152, 794–817 (2001).
[CrossRef]

S. K. Turitsyn, T. Schäfer, N. J. Doran, K. H. Spatschek, and V. K. Mezentsev, “Pathaveraged chirped optical soliton in dispersion-managed fiber communication lines,” Opt. Commun. 163, 122–158 (1999).
[CrossRef]

I. Gabitov and S. K. Turitsyn, “Breathing solitons in optical fiber links,” JETP Lett. 63, 861–866 (1996).
[CrossRef]

I. Gabitov and S. K. Turitsyn, “Averaged pulse dynamics in a cascaded transmission system with passive dispersion compensation,” Opt. Lett. 21, 327–329 (1996).
[CrossRef]

Veng, T.

Villeneuve, A.

Wei, X.

Weinstein, C.

C. Weinstein, “Fiber design improves long-haul performance,” Laser Focus World 33, 215 (1997).

Xie, C.

Zharnitsky, V.

D. E. Pelinovsky and V. Zharnitsky, “Averaging of dispersion-managed solitons: existence and stability,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 63, 745–776 (2003).
[CrossRef]

V. Zharnitsky, E. Grenier, C. K. R. T. Jones, and S. K. Turitsyn, “Stabilizing effects of dispersion management,” Physica D 152, 794–817 (2001).
[CrossRef]

Electron. Lett.

N. J. Smith, F. M. Knox, N. J. Doran, K. J. Blow, and I. Bennion, “Enhanced power solitons in optical fiber transmission line,” Electron. Lett. 32, 54–55 (1996).
[CrossRef]

T. Lakoba and D. J. Kaup, “Shape of the stationary pulse in the strong dispersion management regime,” Electron. Lett. 34, 1124–1125 (1998).
[CrossRef]

IEEE Photonics Technol. Lett.

C. Kurtzke, “Suppression of fiber nonlinearities by appropriate dispersion management,” IEEE Photonics Technol. Lett. 5, 1250–1253 (1993).
[CrossRef]

A. R. Chraplyvy, A. H. Gnauck, R. W. Tkach, and R. M. Derosier, “8×10 Gb/s transmission through 280 km of dispersion-managed fiber,” IEEE Photonics Technol. Lett. 5, 1233–1235 (1993).
[CrossRef]

JETP Lett.

I. Gabitov and S. K. Turitsyn, “Breathing solitons in optical fiber links,” JETP Lett. 63, 861–866 (1996).
[CrossRef]

P. M. Lushnikov, “On the boundary of the dispersion-managed soliton existence,” JETP Lett. 72, 111–114 (2000).
[CrossRef]

Laser Focus World

C. Weinstein, “Fiber design improves long-haul performance,” Laser Focus World 33, 215 (1997).

Opt. Commun.

S. K. Turitsyn, T. Schäfer, N. J. Doran, K. H. Spatschek, and V. K. Mezentsev, “Pathaveraged chirped optical soliton in dispersion-managed fiber communication lines,” Opt. Commun. 163, 122–158 (1999).
[CrossRef]

Opt. Lett.

L. F. Mollenauer, P. V. Mamyshev, J. Gripp, M. J. Neubelt, N. Mamysheva, L. Grüner-Nielsen, and T. Veng, “Demonstration of massive wavelength-division multiplexing over transoceanic distances by use of dispersion-managed solitons,” Opt. Lett. 25, 704–706 (2000).
[CrossRef]

C. Lin, H. Kogelnik, and L. G. Cohen, “Optical-pulse equalization of low-dispersion transmission in single-mode fibers in the 1.3–1.7-μm spectral region,” Opt. Lett. 5, 476–478 (1980).
[CrossRef] [PubMed]

S. Kumar and A. Hasegawa, “Quasi-soliton propagation in dispersion-managed optical fibers,” Opt. Lett. 22, 372–374 (1997).
[CrossRef] [PubMed]

P. V. Mamyshev and N. A. Mamysheva, “Pulse-overlapped dispersion-managed data transmission and intrachannel four-wave mixing,” Opt. Lett. 24, 1454–1456 (1999).
[CrossRef]

I. Gabitov and S. K. Turitsyn, “Averaged pulse dynamics in a cascaded transmission system with passive dispersion compensation,” Opt. Lett. 21, 327–329 (1996).
[CrossRef]

C. Paré, A. Villeneuve, P.-A. Bélanger, and N. J. Doran, “Compensating for dispersion and the nonlinear Kerr effect without phase conjugation,” Opt. Lett. 21, 459–461 (1996).
[CrossRef] [PubMed]

P. M. Lushnikov, “Dispersion-managed soliton in optical fibers with zero average dispersion,” Opt. Lett. 25, 1144–1146 (2000).
[CrossRef]

M. J. Ablowitz and G. Biondini, “Multiscale pulse dynamics in communication systems with strong dispersion management,” Opt. Lett. 23, 1668–1670 (1998).
[CrossRef]

P. M. Lushnikov, “Dispersion-managed soliton in a strong dispersion map limit,” Opt. Lett. 26, 1535–1537 (2001).
[CrossRef]

I. R. Gabitov and P. M. Lushnikov, “Nonlinearity management in a dispersion-managed system,” Opt. Lett. 27, 113–115 (2002).
[CrossRef]

P. M. Lushnikov, “Fully parallel algorithm for simulating wavelength-division-multiplexed optical fiber systems,” Opt. Lett. 27, 939–941 (2002).
[CrossRef]

X. Wei, X. Liu, C. Xie, and L. F. Mollenauer, “Reduction of collision-induced timing jitter in dense wavelength-division multiplexing by the use of periodic-group-delay dispersion compensators,” Opt. Lett. 28, 983–985 (2003).
[CrossRef] [PubMed]

L. F. Mollenauer, A. Grant, X. Liu, X. Wei, C. Xie, and I. Kang, “Experimental test of dense wavelength-division multiplexing using novel, periodic-group-delay-complemented dispersion compensation and dispersion-managed solitons,” Opt. Lett. 28, 2043–2045 (2003).
[CrossRef] [PubMed]

Phys. Rev. E

S. B. Medvedev, O. V. Shtyrina, S. L. Musher, and M. P. Fedoruk, “Path-averaged optical soliton in double-periodic dispersion-managed systems,” Phys. Rev. E 66, 066607 (2002).
[CrossRef]

D. E. Pelinovsky, “Instabilities of dispersion-managed solitons in the normal dispersion regime,” Phys. Rev. E 62, 4283–4293 (2000).
[CrossRef]

Physica D

V. Zharnitsky, E. Grenier, C. K. R. T. Jones, and S. K. Turitsyn, “Stabilizing effects of dispersion management,” Physica D 152, 794–817 (2001).
[CrossRef]

M. Kunze, “The singular perturbation limit of a variational problem from nonlinear fiber optics,” Physica D 180, 108–114 (2003).
[CrossRef]

SIAM (Soc. Ind. Appl. Math.) J. Appl. Math.

D. E. Pelinovsky and V. Zharnitsky, “Averaging of dispersion-managed solitons: existence and stability,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 63, 745–776 (2003).
[CrossRef]

Other

A. Gajadharsingh and P.-A. Bélanger, “Dispersion-managed solitons as the interference of chirped complex conjugate pulses,” in Nonlinear Guided Waves and their Applications, 2004 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 2004), paper MC7.

B. Biotteau, J. P. Hamaide, F. Pitel, O. Audoin, P. Nouchi, and P. Sansonetti, “Enhancement of soliton system performance by use of new large effective area fibres,” Electron. Lett. 31, 2026–2027 (1995).

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Figures (3)

Fig. 1
Fig. 1

Dispersion d(z) as a function of propagation distance z.

Fig. 2
Fig. 2

DM soliton shape (curve 1) versus Eq. (4) (curve 2) for (a) d=0, s=1, λ=1 and (b) for d=0.1, s=1, λ=1. A(t) is an even function.

Fig. 3
Fig. 3

Integration plane (t1, t2) for rhs of Eq. (3) with t>0. The characteristic layer is located along A,B inside the dotted curve.

Equations (24)

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

iAZ-12β2(Z)ATT+σ(Z)|A|2A=0,
iuz+d(z)utt+c(z)|u|2u=0,
iψˆz(ω)-ω2dψˆ+c(z)(2π)2  exp-iΔL1/2zd˜(z)dz
×ψˆ(ω1)ψˆ(ω2)ψˆ*(ω3)δ(ω1+ω2-ω3-ω)×dω1dω2dω3,
iψˆz(ω)-ω2dψˆ+c0(2π)2  sinsΔ2sΔ2ψˆ(ω1)ψˆ(ω2)
×ψˆ*(ω3)δ(ω1+ω2-ω3-ω)dω1dω2dω3,
c0=(c1L1+c2L2)/L,
iψz+dψtt-12πs Cit1t2sψ(t1+t)ψ(t2+t)×ψ*(t1+t2+t)dt1dt2=0,
-λA+dAtt=12πs r(t1, t2, t)dt1dt2,
r(t1, t2, t)=Cit1t2sA(t1+t)A(t2+t)×A(t1+t2+t).
AGauss=p exp-β2t2,
Aasymp(t)=f(t)cos{t2[a0+a(t)]}exp(-b|t|),
Ci(x)=sin(x)/x+O(1/x2),
A(t1+t)A(t2+t)A(t1+t2+t)exp[-b(|t1+t|+|t2+t|+|t1+t2+t|)]
a0=1/(2s),
a(t)=a1/|t|+a2/t2+O(1/|t|3)
ξ=t+t1+t2,
η=t1-t2,
[λg(d)+dt2/s2]f(t)cost22s+a1|t|+a2×exp(-b|t|)= dξ4π0tdηA(ξ)ft-η2ft+η2(η-t)(η+t)×sin[[ξ2+t2]/2s+a1(ξ+|t|)+2a2]×exp(-bξ-b|t|),
[λg(d)+dt2/s2]f(t)=C10tdη˜ f(t-η˜)f(η˜)(t-η˜)η˜,
C1= dξ8πiA(ξ)exp[i(ξ2/2s+a1ξ+a2)-bξ],
C1=Re(C1),
λg(d)df˜pdp+d1s2d3f˜pdp3=-C1f˜p2,
f(t)=λ|t|C1ford=0,f(t)=30λ|t|3C1ford0.

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