Abstract

A detailed analysis of the dynamics of classical (sech) and dispersion-managed solitons is developed for a channel guided by filters of different types, with emphasis on comparison of results produced by etalon and notch filters. By means of analytical approximations, based on balance equations for the soliton’s energy and momentum, we demonstrate that finite-width notch filters can be as efficient as etalon filters, and attainable filter efficiency is higher for dispersion-managed solitons than for classical solitons. We also show that values for filter parameters can be found that maximize the filter’s restoring force.

© 2002 Optical Society of America

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References

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  1. E. A. Golovchenko, A. N. Pilipetskii, and C. R. Menyuk, “Collision-induced timing jitter reduction by periodic dispersion management in soliton wavelength-division-multiplexed transmission,” Electron. Lett. 33, 735–737 (1997).
    [CrossRef]
  2. P. V. Mamyshev and L. F. Mollenauer, “Soliton collisions in wavelength-division-multiplexed dispersion-managed systems,” Opt. Lett. 24, 448–450 (1999).
    [CrossRef]
  3. A. M. Niculae, W. Forysiak, A. Gloag, J. H. B. Nijhof, and N. J. Doran, “Soliton collisions with wavelength-division-multiplexed systems with strong dispersion management,” Opt. Lett. 23, 1354–1356 (1998).
    [CrossRef]
  4. H. Sugahara, H. Kato, and Y. Kodama, “Maximum reductions of collision induced frequency shift in soliton wavelength-division-multiplexed systems with dispersion compensation,” Electron. Lett. 33, 1065–1066 (1997).
    [CrossRef]
  5. D. J. Kaup, B. Malomed, and J. Yang, “Interchannel pulse collision in a wavelength-division-multiplexed system with strong dispersion management,” Opt. Lett. 23, 1600–1602 (1998).
    [CrossRef]
  6. S. Wabnitz, “Role of the filter spectral profile in the control of soliton transmissions,” Opt. Commun. 130, 89–96 (1996).
    [CrossRef]
  7. S. Wabnitz, “Stabilization of sliding-filter soliton wavelength-division-multiplexed transmissions by dispersion compensating fibers,” Opt. Lett. 21, 638–640 (1996).
    [CrossRef] [PubMed]
  8. Y. Chen and H. A. Haus, “Collisions in dispersion-managed soliton propagation,” Opt. Lett. 24, 217–219 (1999).
    [CrossRef]
  9. L. F. Mollenauer, J. Gordon, and S. Evangelides, “The sliding-frequency guiding filter: an improved form of soliton jitter control,” Opt. Lett. 17, 1575–1577 (1992).
    [CrossRef] [PubMed]
  10. M. Hanna, H. Porte, J.-P. Goedgebuer, and W. T. Rhodes, “Soliton optical phase control by use of in-line filters,” Opt. Lett. 24, 732–734 (1999).
    [CrossRef]
  11. A. Mecozzi, “Soliton transmission control by Butterworth filters,” Opt. Lett. 20, 1859–1861 (1995).
    [CrossRef] [PubMed]
  12. Y. Kodama and A. Hasegawa, “Generation of asymptotically stable optical solitons and suppression of the Gordon–Haus effect,” Opt. Lett. 17, 31–33 (1992).
    [CrossRef] [PubMed]
  13. P. V. Mamyshev and L. F. Mollenauer, “Wavelength-division-multiplexing channel energy self-equalization in a soliton transmission line by guiding filters,” Opt. Lett. 21, 1658–1660 (1996).
    [CrossRef] [PubMed]
  14. A. Berntson and B. A. Malomed, “Dispersion management with filtering,” Opt. Lett. 24, 507–509 (1999).
    [CrossRef]
  15. L. F. Mollenauer, P. V. Mamyshev, and J. P. Gordon, “Effect of guiding filters on the behavior of dispersion-managed solitons,” Opt. Lett. 24, 220–222 (1999).
    [CrossRef]
  16. M. Matsumoto, “Analysis of filter control of dispersion-managed soliton transmission,” J. Opt. Soc. Am. B 15, 2831–2837 (1998).
    [CrossRef]
  17. M. Matsumoto, “Instability of dispersion-managed solitons in a system with filtering,” Opt. Lett. 23, 1901–1903 (1998).
    [CrossRef]
  18. B. Malomed, G. D. Peng, and P. L. Chu, “Soliton wavelength-division multiplexing system with channel-isolating notch filters,” Opt. Lett. 24, 1100–1102 (1999).
    [CrossRef]
  19. B. A. Malomed, A. Docherty, P. L. Chu, and G. D. Peng, “Dense wavelength-division-multiplexed soliton systems using channel-isolating notch filters (soliton rail),” in Massive Wavelength-Division-Multiplexed and Time-Division-Multiplexed soliton transmission systems, A. Hasegawa, ed. (Kluwer Academic, Boston, 2000); pp. 411–424.
  20. S. Orlov, A. Yariv, and S. V. Essen, “Coupled-mode analysis of fiber-optic add-drop filters for dense wavelength division multiplexing,” Opt. Lett. 22, 688–690 (1997).
    [CrossRef] [PubMed]
  21. X. J. Gu, “Wavelength-division-multiplexed isolation fiber filter and light source using cascading long-period fiber gratings,” Opt. Lett. 23, 509–511 (1998).
    [CrossRef]
  22. M. J. Ablowitz, G. Biondini, S. Chakravarty, and R. Horne, “A comparison between lumped and distributed filter models in wavelength-division multiplexed soliton systems,” Opt. Commun. 172, 211–227 (1999).
    [CrossRef]
  23. T. I. Lakoba, J. Yang, D. J. Kaup, and B. A. Malomed, “Conditions for stationary pulse propagation in the strong dispersion management regime,” Opt. Commun. 149, 366–375 (1998).
    [CrossRef]
  24. M. J. Ablowitz and G. Biondini, “Multiscale pulse dynamics in communication systems with strong dispersion management,” Opt. Lett. 23, 1668–1670 (1998)
    [CrossRef]
  25. T. S. Yang and W. L. Kath, “Analysis of enhanced-power solitons in dispersion-managed optical fibers,” Opt. Lett. 22, 985–987 (1997).
    [CrossRef] [PubMed]
  26. V. Zharnitsky, E. Grenier, S. K. Turitsyn, C. Jones, and J. S. Hesthaven, “Ground states of dispersion-managed nonlinear Schrodinger equation,” Phys. Rev. E 62, 7358–7364 (2000).
    [CrossRef]
  27. J. H. B. Nijhof, W. Forysiak, and N. J. Doran, “The averaging method for finding exactly periodic dispersion-managed solitons,” IEEE J. Sel. Top. Quantum Electron. 6, 330–336 (2000).
    [CrossRef]
  28. K. J. Blow, N. J. Doran, and D. Wood, “Suppression of the soliton self-frequency shift by bandwidth-limited amplification,” J. Opt. Soc. Am. B 5, 1301–1304 (1988).
    [CrossRef]

2000 (2)

V. Zharnitsky, E. Grenier, S. K. Turitsyn, C. Jones, and J. S. Hesthaven, “Ground states of dispersion-managed nonlinear Schrodinger equation,” Phys. Rev. E 62, 7358–7364 (2000).
[CrossRef]

J. H. B. Nijhof, W. Forysiak, and N. J. Doran, “The averaging method for finding exactly periodic dispersion-managed solitons,” IEEE J. Sel. Top. Quantum Electron. 6, 330–336 (2000).
[CrossRef]

1999 (7)

1998 (7)

1997 (4)

S. Orlov, A. Yariv, and S. V. Essen, “Coupled-mode analysis of fiber-optic add-drop filters for dense wavelength division multiplexing,” Opt. Lett. 22, 688–690 (1997).
[CrossRef] [PubMed]

T. S. Yang and W. L. Kath, “Analysis of enhanced-power solitons in dispersion-managed optical fibers,” Opt. Lett. 22, 985–987 (1997).
[CrossRef] [PubMed]

E. A. Golovchenko, A. N. Pilipetskii, and C. R. Menyuk, “Collision-induced timing jitter reduction by periodic dispersion management in soliton wavelength-division-multiplexed transmission,” Electron. Lett. 33, 735–737 (1997).
[CrossRef]

H. Sugahara, H. Kato, and Y. Kodama, “Maximum reductions of collision induced frequency shift in soliton wavelength-division-multiplexed systems with dispersion compensation,” Electron. Lett. 33, 1065–1066 (1997).
[CrossRef]

1996 (3)

1995 (1)

1992 (2)

1988 (1)

Ablowitz, M. J.

M. J. Ablowitz, G. Biondini, S. Chakravarty, and R. Horne, “A comparison between lumped and distributed filter models in wavelength-division multiplexed soliton systems,” Opt. Commun. 172, 211–227 (1999).
[CrossRef]

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

Berntson, A.

Biondini, G.

M. J. Ablowitz, G. Biondini, S. Chakravarty, and R. Horne, “A comparison between lumped and distributed filter models in wavelength-division multiplexed soliton systems,” Opt. Commun. 172, 211–227 (1999).
[CrossRef]

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

Blow, K. J.

Chakravarty, S.

M. J. Ablowitz, G. Biondini, S. Chakravarty, and R. Horne, “A comparison between lumped and distributed filter models in wavelength-division multiplexed soliton systems,” Opt. Commun. 172, 211–227 (1999).
[CrossRef]

Chen, Y.

Chu, P. L.

Doran, N. J.

Essen, S. V.

Evangelides, S.

Forysiak, W.

J. H. B. Nijhof, W. Forysiak, and N. J. Doran, “The averaging method for finding exactly periodic dispersion-managed solitons,” IEEE J. Sel. Top. Quantum Electron. 6, 330–336 (2000).
[CrossRef]

A. M. Niculae, W. Forysiak, A. Gloag, J. H. B. Nijhof, and N. J. Doran, “Soliton collisions with wavelength-division-multiplexed systems with strong dispersion management,” Opt. Lett. 23, 1354–1356 (1998).
[CrossRef]

Gloag, A.

Goedgebuer, J.-P.

Golovchenko, E. A.

E. A. Golovchenko, A. N. Pilipetskii, and C. R. Menyuk, “Collision-induced timing jitter reduction by periodic dispersion management in soliton wavelength-division-multiplexed transmission,” Electron. Lett. 33, 735–737 (1997).
[CrossRef]

Gordon, J.

Gordon, J. P.

Grenier, E.

V. Zharnitsky, E. Grenier, S. K. Turitsyn, C. Jones, and J. S. Hesthaven, “Ground states of dispersion-managed nonlinear Schrodinger equation,” Phys. Rev. E 62, 7358–7364 (2000).
[CrossRef]

Gu, X. J.

Hanna, M.

Hasegawa, A.

Haus, H. A.

Hesthaven, J. S.

V. Zharnitsky, E. Grenier, S. K. Turitsyn, C. Jones, and J. S. Hesthaven, “Ground states of dispersion-managed nonlinear Schrodinger equation,” Phys. Rev. E 62, 7358–7364 (2000).
[CrossRef]

Horne, R.

M. J. Ablowitz, G. Biondini, S. Chakravarty, and R. Horne, “A comparison between lumped and distributed filter models in wavelength-division multiplexed soliton systems,” Opt. Commun. 172, 211–227 (1999).
[CrossRef]

Jones, C.

V. Zharnitsky, E. Grenier, S. K. Turitsyn, C. Jones, and J. S. Hesthaven, “Ground states of dispersion-managed nonlinear Schrodinger equation,” Phys. Rev. E 62, 7358–7364 (2000).
[CrossRef]

Kath, W. L.

Kato, H.

H. Sugahara, H. Kato, and Y. Kodama, “Maximum reductions of collision induced frequency shift in soliton wavelength-division-multiplexed systems with dispersion compensation,” Electron. Lett. 33, 1065–1066 (1997).
[CrossRef]

Kaup, D. J.

D. J. Kaup, B. Malomed, and J. Yang, “Interchannel pulse collision in a wavelength-division-multiplexed system with strong dispersion management,” Opt. Lett. 23, 1600–1602 (1998).
[CrossRef]

T. I. Lakoba, J. Yang, D. J. Kaup, and B. A. Malomed, “Conditions for stationary pulse propagation in the strong dispersion management regime,” Opt. Commun. 149, 366–375 (1998).
[CrossRef]

Kodama, Y.

H. Sugahara, H. Kato, and Y. Kodama, “Maximum reductions of collision induced frequency shift in soliton wavelength-division-multiplexed systems with dispersion compensation,” Electron. Lett. 33, 1065–1066 (1997).
[CrossRef]

Y. Kodama and A. Hasegawa, “Generation of asymptotically stable optical solitons and suppression of the Gordon–Haus effect,” Opt. Lett. 17, 31–33 (1992).
[CrossRef] [PubMed]

Lakoba, T. I.

T. I. Lakoba, J. Yang, D. J. Kaup, and B. A. Malomed, “Conditions for stationary pulse propagation in the strong dispersion management regime,” Opt. Commun. 149, 366–375 (1998).
[CrossRef]

Malomed, B.

Malomed, B. A.

A. Berntson and B. A. Malomed, “Dispersion management with filtering,” Opt. Lett. 24, 507–509 (1999).
[CrossRef]

T. I. Lakoba, J. Yang, D. J. Kaup, and B. A. Malomed, “Conditions for stationary pulse propagation in the strong dispersion management regime,” Opt. Commun. 149, 366–375 (1998).
[CrossRef]

Mamyshev, P. V.

Matsumoto, M.

Mecozzi, A.

Menyuk, C. R.

E. A. Golovchenko, A. N. Pilipetskii, and C. R. Menyuk, “Collision-induced timing jitter reduction by periodic dispersion management in soliton wavelength-division-multiplexed transmission,” Electron. Lett. 33, 735–737 (1997).
[CrossRef]

Mollenauer, L. F.

Niculae, A. M.

Nijhof, J. H. B.

J. H. B. Nijhof, W. Forysiak, and N. J. Doran, “The averaging method for finding exactly periodic dispersion-managed solitons,” IEEE J. Sel. Top. Quantum Electron. 6, 330–336 (2000).
[CrossRef]

A. M. Niculae, W. Forysiak, A. Gloag, J. H. B. Nijhof, and N. J. Doran, “Soliton collisions with wavelength-division-multiplexed systems with strong dispersion management,” Opt. Lett. 23, 1354–1356 (1998).
[CrossRef]

Orlov, S.

Peng, G. D.

Pilipetskii, A. N.

E. A. Golovchenko, A. N. Pilipetskii, and C. R. Menyuk, “Collision-induced timing jitter reduction by periodic dispersion management in soliton wavelength-division-multiplexed transmission,” Electron. Lett. 33, 735–737 (1997).
[CrossRef]

Porte, H.

Rhodes, W. T.

Sugahara, H.

H. Sugahara, H. Kato, and Y. Kodama, “Maximum reductions of collision induced frequency shift in soliton wavelength-division-multiplexed systems with dispersion compensation,” Electron. Lett. 33, 1065–1066 (1997).
[CrossRef]

Turitsyn, S. K.

V. Zharnitsky, E. Grenier, S. K. Turitsyn, C. Jones, and J. S. Hesthaven, “Ground states of dispersion-managed nonlinear Schrodinger equation,” Phys. Rev. E 62, 7358–7364 (2000).
[CrossRef]

Wabnitz, S.

Wood, D.

Yang, J.

D. J. Kaup, B. Malomed, and J. Yang, “Interchannel pulse collision in a wavelength-division-multiplexed system with strong dispersion management,” Opt. Lett. 23, 1600–1602 (1998).
[CrossRef]

T. I. Lakoba, J. Yang, D. J. Kaup, and B. A. Malomed, “Conditions for stationary pulse propagation in the strong dispersion management regime,” Opt. Commun. 149, 366–375 (1998).
[CrossRef]

Yang, T. S.

Yariv, A.

Zharnitsky, V.

V. Zharnitsky, E. Grenier, S. K. Turitsyn, C. Jones, and J. S. Hesthaven, “Ground states of dispersion-managed nonlinear Schrodinger equation,” Phys. Rev. E 62, 7358–7364 (2000).
[CrossRef]

Electron. Lett. (2)

E. A. Golovchenko, A. N. Pilipetskii, and C. R. Menyuk, “Collision-induced timing jitter reduction by periodic dispersion management in soliton wavelength-division-multiplexed transmission,” Electron. Lett. 33, 735–737 (1997).
[CrossRef]

H. Sugahara, H. Kato, and Y. Kodama, “Maximum reductions of collision induced frequency shift in soliton wavelength-division-multiplexed systems with dispersion compensation,” Electron. Lett. 33, 1065–1066 (1997).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

J. H. B. Nijhof, W. Forysiak, and N. J. Doran, “The averaging method for finding exactly periodic dispersion-managed solitons,” IEEE J. Sel. Top. Quantum Electron. 6, 330–336 (2000).
[CrossRef]

J. Opt. Soc. Am. B (2)

Opt. Commun. (3)

M. J. Ablowitz, G. Biondini, S. Chakravarty, and R. Horne, “A comparison between lumped and distributed filter models in wavelength-division multiplexed soliton systems,” Opt. Commun. 172, 211–227 (1999).
[CrossRef]

T. I. Lakoba, J. Yang, D. J. Kaup, and B. A. Malomed, “Conditions for stationary pulse propagation in the strong dispersion management regime,” Opt. Commun. 149, 366–375 (1998).
[CrossRef]

S. Wabnitz, “Role of the filter spectral profile in the control of soliton transmissions,” Opt. Commun. 130, 89–96 (1996).
[CrossRef]

Opt. Lett. (18)

Y. Kodama and A. Hasegawa, “Generation of asymptotically stable optical solitons and suppression of the Gordon–Haus effect,” Opt. Lett. 17, 31–33 (1992).
[CrossRef] [PubMed]

L. F. Mollenauer, J. Gordon, and S. Evangelides, “The sliding-frequency guiding filter: an improved form of soliton jitter control,” Opt. Lett. 17, 1575–1577 (1992).
[CrossRef] [PubMed]

A. Mecozzi, “Soliton transmission control by Butterworth filters,” Opt. Lett. 20, 1859–1861 (1995).
[CrossRef] [PubMed]

S. Orlov, A. Yariv, and S. V. Essen, “Coupled-mode analysis of fiber-optic add-drop filters for dense wavelength division multiplexing,” Opt. Lett. 22, 688–690 (1997).
[CrossRef] [PubMed]

T. S. Yang and W. L. Kath, “Analysis of enhanced-power solitons in dispersion-managed optical fibers,” Opt. Lett. 22, 985–987 (1997).
[CrossRef] [PubMed]

X. J. Gu, “Wavelength-division-multiplexed isolation fiber filter and light source using cascading long-period fiber gratings,” Opt. Lett. 23, 509–511 (1998).
[CrossRef]

A. M. Niculae, W. Forysiak, A. Gloag, J. H. B. Nijhof, and N. J. Doran, “Soliton collisions with wavelength-division-multiplexed systems with strong dispersion management,” Opt. Lett. 23, 1354–1356 (1998).
[CrossRef]

D. J. Kaup, B. Malomed, and J. Yang, “Interchannel pulse collision in a wavelength-division-multiplexed system with strong dispersion management,” Opt. Lett. 23, 1600–1602 (1998).
[CrossRef]

M. Matsumoto, “Instability of dispersion-managed solitons in a system with filtering,” Opt. Lett. 23, 1901–1903 (1998).
[CrossRef]

Y. Chen and H. A. Haus, “Collisions in dispersion-managed soliton propagation,” Opt. Lett. 24, 217–219 (1999).
[CrossRef]

L. F. Mollenauer, P. V. Mamyshev, and J. P. Gordon, “Effect of guiding filters on the behavior of dispersion-managed solitons,” Opt. Lett. 24, 220–222 (1999).
[CrossRef]

P. V. Mamyshev and L. F. Mollenauer, “Soliton collisions in wavelength-division-multiplexed dispersion-managed systems,” Opt. Lett. 24, 448–450 (1999).
[CrossRef]

A. Berntson and B. A. Malomed, “Dispersion management with filtering,” Opt. Lett. 24, 507–509 (1999).
[CrossRef]

M. Hanna, H. Porte, J.-P. Goedgebuer, and W. T. Rhodes, “Soliton optical phase control by use of in-line filters,” Opt. Lett. 24, 732–734 (1999).
[CrossRef]

B. Malomed, G. D. Peng, and P. L. Chu, “Soliton wavelength-division multiplexing system with channel-isolating notch filters,” Opt. Lett. 24, 1100–1102 (1999).
[CrossRef]

S. Wabnitz, “Stabilization of sliding-filter soliton wavelength-division-multiplexed transmissions by dispersion compensating fibers,” Opt. Lett. 21, 638–640 (1996).
[CrossRef] [PubMed]

P. V. Mamyshev and L. F. Mollenauer, “Wavelength-division-multiplexing channel energy self-equalization in a soliton transmission line by guiding filters,” Opt. Lett. 21, 1658–1660 (1996).
[CrossRef] [PubMed]

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

Phys. Rev. E (1)

V. Zharnitsky, E. Grenier, S. K. Turitsyn, C. Jones, and J. S. Hesthaven, “Ground states of dispersion-managed nonlinear Schrodinger equation,” Phys. Rev. E 62, 7358–7364 (2000).
[CrossRef]

Other (1)

B. A. Malomed, A. Docherty, P. L. Chu, and G. D. Peng, “Dense wavelength-division-multiplexed soliton systems using channel-isolating notch filters (soliton rail),” in Massive Wavelength-Division-Multiplexed and Time-Division-Multiplexed soliton transmission systems, A. Hasegawa, ed. (Kluwer Academic, Boston, 2000); pp. 411–424.

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Figures (7)

Fig. 1
Fig. 1

Relative energy difference between a numerically exact dispersion-managed soliton (with a map strength S=2) and the best-fit Gaussian approximation to it, versus the net energy of the numerically found soliton.

Fig. 2
Fig. 2

Comparison between amplitude a and inverse width b of the Gaussian pulse of Eq. (8) with etalon filterering (=0.1), (i) from numerical solution of the analytically derived Eqs. (14), (ii) from numerical solution of the energy-evolution Eq. (11) with a(E) and b(E) taken from numerically generated stationary DM solitons, and (iii) from direct numerical simulations of the NLSE (1).

Fig. 3
Fig. 3

Optimal values of ν (a) and ω0 (b) versus for classical and DM solitons, with α0=0.05 for Gaussian and square-notch filters, as obtained from Eqs. (5), (6) and Eqs. (11), (12) with integrals calculated numerically.

Fig. 4
Fig. 4

Value of H versus for (a) classical solitons and (b) DM solitons as obtained from Eqs. (20) and (21) and from direct numerical simulations with α0=0.05.

Fig. 5
Fig. 5

Value of H versus α0 for (a) the etalon and (b) Butterworth filters both numerically and analytically.

Fig. 6
Fig. 6

Direct test of frequency-jitter suppression. The plots show the temporal offset for (a) classical solitons and (b) DM solitons versus the propagation distance. Note that although the filter effectiveness is the same for both classical and DM solitons (H=4α0), the frequency jitter translates to timing jitter differently for both, the DM solitons having significantly less consequent timing jitter.

Fig. 7
Fig. 7

Final values of the timing jitter in the lumped version of the dispersion-managed model with the Gaussian notch filters versus the normalized filter spacing za.

Equations (30)

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i uz+D(z)2utt+|u|2u=if(u, t)+iα0u,
i uˆz-D(z)2ω2uˆ+i-|u|2u exp(iωt)dt
=iF(ω)yˆ+iα0uˆ,
E-|uˆ(ω)|2dω=-|u(t)|2dt
M-ω|uˆ(ω)|2dω=iπ-(u*ut-uut*)dt
z -|uˆ|2dω=2-R[F(ω)]|uˆ|2dω+2α0-|uˆ|2dω,
z -ω|uˆ|2dω=2-ωR[F(ω)]|uˆ|2dω+2α0-ω|uˆ|2dω.
α0=- -Fr(ω)|uˆ0(ω)|2dω-|uˆ0|2dω.
uˆ(z, ω)=π sech π2η(z)[ω-Ω(z)],
ηz=2αη(z)+π2-Fr(ω)×sech2π2η(z)[ω-Ω(z)]dω,
Ωz=π2η-[ω-Ω(z)]Fr(ω)×sech2π2η(z)[ω-Ω(z)]dω.
α0=-π4η0-Fr(ω)sech2π2η0(ω-Ω0)dω,
|uˆGauss(ω, z)|2=a(z)2 exp{-b(z)[ω-Ω(z)2]},
a2=2D¯Sb 1+(S/b)2sinh-1(S/b),
N-[|uDM(ω)|-a exp(-bω2/2)]2dω.
dEdz=2α0E+2-Fr(ω)a(z)2×exp{-b(z)[ω-Ω(z)]2}dω,
dΩdz=2E(z)-[ω-Ω(z)]Fr(ω)a(z)2×exp{-b(z)[ω-Ω(z)]2}dω.
α0=-b0π -Fr(ω)exp[-b0(ω-Ω0)2]dω,
dadz=2a α+bπ-Fr(ω)exp[-b(ω-Ω)2]dω2-(a/2b)db/da,
dbdz=2a α+bπ -Fr(ω)exp[-b(ω-Ω)2]dω2da/db-a/2b.
FS=1if|ω-ω0(1, 2)|<νs/20otherwise,
FG=m=1,2 exp-1νg(ω-ω0(m))2,
dEdzsech4α0η-π3/2νgsech2π2η(ω0-Ω)+sech2π2η(ω0+Ω),
dΩdzsech π3/2νg2η (Ω-ω0)sech2π2η(ω0-Ω)+(Ω+ω0)sech2π2η(ω0+Ω).
dEdzDM2α0E-2πνga2{exp[-b(ω0-Ω)2]+exp[-b(ω0+Ω)2]},
dΩdzDM-2Ea2πνg{(ω0-Ω)exp[-b(ω0-Ω)2]+(ω0+Ω)exp[-b(ω0+Ω)2]}.
dΩdz=-H(Ω-Ω0)+O[(Ω-Ω0)2],
H=-π2η-Fr(ω)sech2πω2η πωη tanh πω2η-1dω.
H=-2bπ -Fr(ω)(2ω2b-1)exp(-bω2)dω.
sech=3α0/η,DM=2bα0,

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