Abstract

In the presence of pre-dispersion, an exact solution of nonlinear Schrödinger equation (NLSE) is derived for impulse input. The phase factor of the exact solution is obtained in a closed form using the exponential integral. The nonlinear interaction among periodically placed impulses launched at the input is investigated, and the condition under which these pulses do not exchange energy is examined. It is found that if the complex weights of the impulses at the input have a secant-hyperbolic envelope and a proper chirp factor, they will propagate over long distances without exchanging energy. To describe their interaction, a discrete version of NLSE is derived. The derived equation is a form of discrete self-trapping (DST) equation, which is found to admit fundamental and higher order soliton solutions in the presence of high pre-dispersion. Nonlinear eigenmodes derived here may be useful for description of signal propagation and nonlinear interaction in highly pre-dispersion fiber-optic systems.

© 2014 Optical Society of America

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References

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  1. A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical physics in dispersive dielectric fibers I: anomalous dispersion,” Appl. Phys. Lett. 23(3), 142–144 (1973).
    [Crossref]
  2. V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34(1), 62–69 (1972).
  3. G. L. Lamb, Elements of Soliton Theory (John Wiley & Sons, INc. 1980).
  4. P. G. Drazin and R. S. Johnson, Solitons: An Introduction (Cambridge University Press1989).
    [Crossref]
  5. Y. Xiao, D. N. Maywar, and G. P. Agrawal, “New approach to pulse propagation in nonlinear dispersive optical media,” J. Opt. Soc. Am. B. 29(10), 2958–2963 (2012).
    [Crossref]
  6. E. Ciaramella and E. Forestieri, “Analytical approximation of nonlinear distortions,” IEEE Photon. Technol. Lett. 17(1), 91–93 (2005).
    [Crossref]
  7. R. J. Essiambre, B. Mikkelsen, and G. Raybon, “Intra-channel cross-phase modulation and four-wave mixing in high-speed TDM systems,” Electron. Lett. 35(18), 1576–1578 (1999).
    [Crossref]
  8. A. Mecozzi, C. B. Clausen, and M. Shtaif, “Analysis of intrachannel nonlinear effects in highly dispersed optical pulse transmission,” IEEE Photon. Technol. Lett. 12(4), 392–394 (2000).
    [Crossref]
  9. P. V. Mamyshev and N. A. Mamysheva, “Pulse-overlapped dispersion-managed data transmission and intrachannel four-wave mixing,” Opt. Lett. 24(21), 1454–1456 (1999).
    [Crossref]
  10. S. Kumar, J. C. Mauro, S. Raghavan, and D. Q. Chowdhury, “Intrachannel nonlinear penalties in dispersion-managed transmission systems,” IEEE J. Sel. Top. Quantum Electron. 8(3), 626–631 (2002).
    [Crossref]
  11. S. Kumar and M. J. Deen, Fiber Optic Communications: Fundamentals and Applications (Wiley2014), Chap. 10.
    [Crossref]
  12. A. Hasegawa and Y. Kodama, “Guiding center soliton,” Phys. Rev. Lett. 66, 161–164 (1991).
    [Crossref] [PubMed]
  13. S. J. Savory, “Digital filters for coherent optical receivers,” Opt. Express,  16(2), 804–817 (2008).
    [Crossref] [PubMed]
  14. I. S. Gradshteyn and I. M. Ryzhik, Table of Integral, Series, and Products 6th (Acdemic Press, 2000), Chap. 6.
  15. M. J. Ablowitz and B. Prinari, “Nonlinear Schrödinger systems: continuous and discrete,” Scholarpedia 3(8), 5561 (2008).
    [Crossref]
  16. J. C. Eilbeck and M. Johansson, “The discrete nonlinear Schrödinger equation - 20 years on,” in Proceedings of the third conference on Localization and Energy Transfer in Nonlinear Systems, (San Lorenzo de El Escorial, Madrid, 2003), pp. 44–67.
    [Crossref]
  17. J. C. Eilbeck, P. S. Lomdahl, and A. C. Scott. , “The discrete self-trapping equation,” Physica D. 16, 318–338(1985).
    [Crossref]
  18. S. Turitsyn, M. Sorokina, and S. Derevyanko, “Dispersion-dominated nonlinear fiber-optic channel,” Opt. Lett. 37(14), 2931–2933 (2012).
    [Crossref] [PubMed]
  19. M. Ablowitz and T. Hirooka, “Managing nonlinearity in strongly dispersion-managed optical pulse transmission,” J. Opt. Soc, Am. B. 19(3), 425–439 (2002).
    [Crossref]
  20. Z. Tao, L. Dou, W. Yan, L. Li, T. Hoshida, and J. C. Rasmussen, “Multiplier-free intrachannel nonlinearity compensating algorithm operating at symbol rate,” J. Lightwave Technol. 29(17), 2570–2576 (2011).
    [Crossref]
  21. X. Liang and S. Kumar, “Multi-stage perturbation theory for compensating intra-channel nonlinear impairments in fiber-optic links,” Opt. Express,  22(24), 29733–29745 (2014).
    [Crossref]

2014 (1)

2012 (2)

S. Turitsyn, M. Sorokina, and S. Derevyanko, “Dispersion-dominated nonlinear fiber-optic channel,” Opt. Lett. 37(14), 2931–2933 (2012).
[Crossref] [PubMed]

Y. Xiao, D. N. Maywar, and G. P. Agrawal, “New approach to pulse propagation in nonlinear dispersive optical media,” J. Opt. Soc. Am. B. 29(10), 2958–2963 (2012).
[Crossref]

2011 (1)

2008 (2)

S. J. Savory, “Digital filters for coherent optical receivers,” Opt. Express,  16(2), 804–817 (2008).
[Crossref] [PubMed]

M. J. Ablowitz and B. Prinari, “Nonlinear Schrödinger systems: continuous and discrete,” Scholarpedia 3(8), 5561 (2008).
[Crossref]

2005 (1)

E. Ciaramella and E. Forestieri, “Analytical approximation of nonlinear distortions,” IEEE Photon. Technol. Lett. 17(1), 91–93 (2005).
[Crossref]

2002 (2)

S. Kumar, J. C. Mauro, S. Raghavan, and D. Q. Chowdhury, “Intrachannel nonlinear penalties in dispersion-managed transmission systems,” IEEE J. Sel. Top. Quantum Electron. 8(3), 626–631 (2002).
[Crossref]

M. Ablowitz and T. Hirooka, “Managing nonlinearity in strongly dispersion-managed optical pulse transmission,” J. Opt. Soc, Am. B. 19(3), 425–439 (2002).
[Crossref]

2000 (1)

A. Mecozzi, C. B. Clausen, and M. Shtaif, “Analysis of intrachannel nonlinear effects in highly dispersed optical pulse transmission,” IEEE Photon. Technol. Lett. 12(4), 392–394 (2000).
[Crossref]

1999 (2)

P. V. Mamyshev and N. A. Mamysheva, “Pulse-overlapped dispersion-managed data transmission and intrachannel four-wave mixing,” Opt. Lett. 24(21), 1454–1456 (1999).
[Crossref]

R. J. Essiambre, B. Mikkelsen, and G. Raybon, “Intra-channel cross-phase modulation and four-wave mixing in high-speed TDM systems,” Electron. Lett. 35(18), 1576–1578 (1999).
[Crossref]

1991 (1)

A. Hasegawa and Y. Kodama, “Guiding center soliton,” Phys. Rev. Lett. 66, 161–164 (1991).
[Crossref] [PubMed]

1985 (1)

J. C. Eilbeck, P. S. Lomdahl, and A. C. Scott. , “The discrete self-trapping equation,” Physica D. 16, 318–338(1985).
[Crossref]

1973 (1)

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical physics in dispersive dielectric fibers I: anomalous dispersion,” Appl. Phys. Lett. 23(3), 142–144 (1973).
[Crossref]

1972 (1)

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34(1), 62–69 (1972).

Ablowitz, M.

M. Ablowitz and T. Hirooka, “Managing nonlinearity in strongly dispersion-managed optical pulse transmission,” J. Opt. Soc, Am. B. 19(3), 425–439 (2002).
[Crossref]

Ablowitz, M. J.

M. J. Ablowitz and B. Prinari, “Nonlinear Schrödinger systems: continuous and discrete,” Scholarpedia 3(8), 5561 (2008).
[Crossref]

Agrawal, G. P.

Y. Xiao, D. N. Maywar, and G. P. Agrawal, “New approach to pulse propagation in nonlinear dispersive optical media,” J. Opt. Soc. Am. B. 29(10), 2958–2963 (2012).
[Crossref]

Chowdhury, D. Q.

S. Kumar, J. C. Mauro, S. Raghavan, and D. Q. Chowdhury, “Intrachannel nonlinear penalties in dispersion-managed transmission systems,” IEEE J. Sel. Top. Quantum Electron. 8(3), 626–631 (2002).
[Crossref]

Ciaramella, E.

E. Ciaramella and E. Forestieri, “Analytical approximation of nonlinear distortions,” IEEE Photon. Technol. Lett. 17(1), 91–93 (2005).
[Crossref]

Clausen, C. B.

A. Mecozzi, C. B. Clausen, and M. Shtaif, “Analysis of intrachannel nonlinear effects in highly dispersed optical pulse transmission,” IEEE Photon. Technol. Lett. 12(4), 392–394 (2000).
[Crossref]

Deen, M. J.

S. Kumar and M. J. Deen, Fiber Optic Communications: Fundamentals and Applications (Wiley2014), Chap. 10.
[Crossref]

Derevyanko, S.

Dou, L.

Drazin, P. G.

P. G. Drazin and R. S. Johnson, Solitons: An Introduction (Cambridge University Press1989).
[Crossref]

Eilbeck, J. C.

J. C. Eilbeck, P. S. Lomdahl, and A. C. Scott. , “The discrete self-trapping equation,” Physica D. 16, 318–338(1985).
[Crossref]

J. C. Eilbeck and M. Johansson, “The discrete nonlinear Schrödinger equation - 20 years on,” in Proceedings of the third conference on Localization and Energy Transfer in Nonlinear Systems, (San Lorenzo de El Escorial, Madrid, 2003), pp. 44–67.
[Crossref]

Essiambre, R. J.

R. J. Essiambre, B. Mikkelsen, and G. Raybon, “Intra-channel cross-phase modulation and four-wave mixing in high-speed TDM systems,” Electron. Lett. 35(18), 1576–1578 (1999).
[Crossref]

Forestieri, E.

E. Ciaramella and E. Forestieri, “Analytical approximation of nonlinear distortions,” IEEE Photon. Technol. Lett. 17(1), 91–93 (2005).
[Crossref]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integral, Series, and Products 6th (Acdemic Press, 2000), Chap. 6.

Hasegawa, A.

A. Hasegawa and Y. Kodama, “Guiding center soliton,” Phys. Rev. Lett. 66, 161–164 (1991).
[Crossref] [PubMed]

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical physics in dispersive dielectric fibers I: anomalous dispersion,” Appl. Phys. Lett. 23(3), 142–144 (1973).
[Crossref]

Hirooka, T.

M. Ablowitz and T. Hirooka, “Managing nonlinearity in strongly dispersion-managed optical pulse transmission,” J. Opt. Soc, Am. B. 19(3), 425–439 (2002).
[Crossref]

Hoshida, T.

Johansson, M.

J. C. Eilbeck and M. Johansson, “The discrete nonlinear Schrödinger equation - 20 years on,” in Proceedings of the third conference on Localization and Energy Transfer in Nonlinear Systems, (San Lorenzo de El Escorial, Madrid, 2003), pp. 44–67.
[Crossref]

Johnson, R. S.

P. G. Drazin and R. S. Johnson, Solitons: An Introduction (Cambridge University Press1989).
[Crossref]

Kodama, Y.

A. Hasegawa and Y. Kodama, “Guiding center soliton,” Phys. Rev. Lett. 66, 161–164 (1991).
[Crossref] [PubMed]

Kumar, S.

X. Liang and S. Kumar, “Multi-stage perturbation theory for compensating intra-channel nonlinear impairments in fiber-optic links,” Opt. Express,  22(24), 29733–29745 (2014).
[Crossref]

S. Kumar, J. C. Mauro, S. Raghavan, and D. Q. Chowdhury, “Intrachannel nonlinear penalties in dispersion-managed transmission systems,” IEEE J. Sel. Top. Quantum Electron. 8(3), 626–631 (2002).
[Crossref]

S. Kumar and M. J. Deen, Fiber Optic Communications: Fundamentals and Applications (Wiley2014), Chap. 10.
[Crossref]

Lamb, G. L.

G. L. Lamb, Elements of Soliton Theory (John Wiley & Sons, INc. 1980).

Li, L.

Liang, X.

Lomdahl, P. S.

J. C. Eilbeck, P. S. Lomdahl, and A. C. Scott. , “The discrete self-trapping equation,” Physica D. 16, 318–338(1985).
[Crossref]

Mamyshev, P. V.

Mamysheva, N. A.

Mauro, J. C.

S. Kumar, J. C. Mauro, S. Raghavan, and D. Q. Chowdhury, “Intrachannel nonlinear penalties in dispersion-managed transmission systems,” IEEE J. Sel. Top. Quantum Electron. 8(3), 626–631 (2002).
[Crossref]

Maywar, D. N.

Y. Xiao, D. N. Maywar, and G. P. Agrawal, “New approach to pulse propagation in nonlinear dispersive optical media,” J. Opt. Soc. Am. B. 29(10), 2958–2963 (2012).
[Crossref]

Mecozzi, A.

A. Mecozzi, C. B. Clausen, and M. Shtaif, “Analysis of intrachannel nonlinear effects in highly dispersed optical pulse transmission,” IEEE Photon. Technol. Lett. 12(4), 392–394 (2000).
[Crossref]

Mikkelsen, B.

R. J. Essiambre, B. Mikkelsen, and G. Raybon, “Intra-channel cross-phase modulation and four-wave mixing in high-speed TDM systems,” Electron. Lett. 35(18), 1576–1578 (1999).
[Crossref]

Prinari, B.

M. J. Ablowitz and B. Prinari, “Nonlinear Schrödinger systems: continuous and discrete,” Scholarpedia 3(8), 5561 (2008).
[Crossref]

Raghavan, S.

S. Kumar, J. C. Mauro, S. Raghavan, and D. Q. Chowdhury, “Intrachannel nonlinear penalties in dispersion-managed transmission systems,” IEEE J. Sel. Top. Quantum Electron. 8(3), 626–631 (2002).
[Crossref]

Rasmussen, J. C.

Raybon, G.

R. J. Essiambre, B. Mikkelsen, and G. Raybon, “Intra-channel cross-phase modulation and four-wave mixing in high-speed TDM systems,” Electron. Lett. 35(18), 1576–1578 (1999).
[Crossref]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integral, Series, and Products 6th (Acdemic Press, 2000), Chap. 6.

Savory, S. J.

Scott., A. C.

J. C. Eilbeck, P. S. Lomdahl, and A. C. Scott. , “The discrete self-trapping equation,” Physica D. 16, 318–338(1985).
[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 nonlinear media,” Sov. Phys. JETP 34(1), 62–69 (1972).

Shtaif, M.

A. Mecozzi, C. B. Clausen, and M. Shtaif, “Analysis of intrachannel nonlinear effects in highly dispersed optical pulse transmission,” IEEE Photon. Technol. Lett. 12(4), 392–394 (2000).
[Crossref]

Sorokina, M.

Tao, Z.

Tappert, F.

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical physics in dispersive dielectric fibers I: anomalous dispersion,” Appl. Phys. Lett. 23(3), 142–144 (1973).
[Crossref]

Turitsyn, S.

Xiao, Y.

Y. Xiao, D. N. Maywar, and G. P. Agrawal, “New approach to pulse propagation in nonlinear dispersive optical media,” J. Opt. Soc. Am. B. 29(10), 2958–2963 (2012).
[Crossref]

Yan, W.

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 nonlinear media,” Sov. Phys. JETP 34(1), 62–69 (1972).

Appl. Phys. Lett. (1)

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical physics in dispersive dielectric fibers I: anomalous dispersion,” Appl. Phys. Lett. 23(3), 142–144 (1973).
[Crossref]

Electron. Lett. (1)

R. J. Essiambre, B. Mikkelsen, and G. Raybon, “Intra-channel cross-phase modulation and four-wave mixing in high-speed TDM systems,” Electron. Lett. 35(18), 1576–1578 (1999).
[Crossref]

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

S. Kumar, J. C. Mauro, S. Raghavan, and D. Q. Chowdhury, “Intrachannel nonlinear penalties in dispersion-managed transmission systems,” IEEE J. Sel. Top. Quantum Electron. 8(3), 626–631 (2002).
[Crossref]

IEEE Photon. Technol. Lett. (2)

A. Mecozzi, C. B. Clausen, and M. Shtaif, “Analysis of intrachannel nonlinear effects in highly dispersed optical pulse transmission,” IEEE Photon. Technol. Lett. 12(4), 392–394 (2000).
[Crossref]

E. Ciaramella and E. Forestieri, “Analytical approximation of nonlinear distortions,” IEEE Photon. Technol. Lett. 17(1), 91–93 (2005).
[Crossref]

J. Lightwave Technol. (1)

J. Opt. Soc, Am. B. (1)

M. Ablowitz and T. Hirooka, “Managing nonlinearity in strongly dispersion-managed optical pulse transmission,” J. Opt. Soc, Am. B. 19(3), 425–439 (2002).
[Crossref]

J. Opt. Soc. Am. B. (1)

Y. Xiao, D. N. Maywar, and G. P. Agrawal, “New approach to pulse propagation in nonlinear dispersive optical media,” J. Opt. Soc. Am. B. 29(10), 2958–2963 (2012).
[Crossref]

Opt. Express (2)

Opt. Lett. (2)

Phys. Rev. Lett. (1)

A. Hasegawa and Y. Kodama, “Guiding center soliton,” Phys. Rev. Lett. 66, 161–164 (1991).
[Crossref] [PubMed]

Physica D. (1)

J. C. Eilbeck, P. S. Lomdahl, and A. C. Scott. , “The discrete self-trapping equation,” Physica D. 16, 318–338(1985).
[Crossref]

Scholarpedia (1)

M. J. Ablowitz and B. Prinari, “Nonlinear Schrödinger systems: continuous and discrete,” Scholarpedia 3(8), 5561 (2008).
[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 nonlinear media,” Sov. Phys. JETP 34(1), 62–69 (1972).

Other (5)

G. L. Lamb, Elements of Soliton Theory (John Wiley & Sons, INc. 1980).

P. G. Drazin and R. S. Johnson, Solitons: An Introduction (Cambridge University Press1989).
[Crossref]

J. C. Eilbeck and M. Johansson, “The discrete nonlinear Schrödinger equation - 20 years on,” in Proceedings of the third conference on Localization and Energy Transfer in Nonlinear Systems, (San Lorenzo de El Escorial, Madrid, 2003), pp. 44–67.
[Crossref]

I. S. Gradshteyn and I. M. Ryzhik, Table of Integral, Series, and Products 6th (Acdemic Press, 2000), Chap. 6.

S. Kumar and M. J. Deen, Fiber Optic Communications: Fundamentals and Applications (Wiley2014), Chap. 10.
[Crossref]

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Figures (4)

Fig. 1
Fig. 1 Evolution of m in the transmission fiber, (a) 0 < th, B ˜ 0 = 10 m W ps, (b) 0 = th. B ˜ th = 14.9 m W ps, M = 28, α = 0 km−1, s0 = −1.28 × 104 ps2, γ0 = 1.1 W−1km−1.
Fig. 2
Fig. 2 Evolution of Bm in the transmission fiber, (a) 0 < th, (b) 0 = th. The parameters are the same as in Fig. 1.
Fig. 3
Fig. 3 Comparison of discrete NLSE (Eq. (28)) and continuous NLSE (Eq. (1)). Peak power = 35.5 mw, T = 10 ps, T0 = 1 ps, s0 = −1.28 × 104 ps2, β2+ = −20 ps2/km, γ0 = 1.1 W−1km−1, transmission distance = 240 km.
Fig. 4
Fig. 4 Evolution of second order soliton. B ˜ 0 = 29.8 m W ps. The rest of the parameters are the same as in Fig. 1.

Equations (47)

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i u z β 2 2 2 u t 2 + γ 0 e α z | u | 2 u = 0 ,
u ( t , 0 ) = A δ ( t ) ,
u ( t , z ) = A i 2 π β 2 z e i t 2 2 β 2 z .
u ( t , z ) = A i 2 π β 2 z e i t 2 2 β 2 z + i v ( z ) ,
A i 2 π β 2 z { i [ ( 1 2 ) z 1 + i t 2 2 β 2 z 2 + i d v ( z ) d z ] β 2 2 ( i β 2 z t 2 β 2 2 z 2 ) + γ 0 e α z | A | 2 2 π | β 2 | z } = 0 .
v ( z ) = γ 0 | A | 2 2 π | β 2 | 0 z e α x x d x .
β 2 ( z ) = { β 2 , for z < 0 β 2 + , for z < 0 ,
γ = { 0 , for z < 0 γ 0 , for z < 0 .
u ( t , z ) = A i 2 π s ( z ) e i t 2 2 s ( z ) + i γ 0 | A | 2 2 π θ ( z ) ,
θ ( z ) = 0 z e α x s ( x ) d x .
s ( z ) = s 0 + β 2 + z .
θ ( z ) = e α s 0 / β 2 + [ Ei ( α s ( z ) β 2 + ) Ei ( α s 0 β 2 + ) ] ,
Ei ( x ) = x e t t d t .
u in ( t ) = n = N / 2 N / 2 1 A n δ ( t n T ) ,
u ( t , z ) = n = N / 2 N / 2 1 A n ( z ) e i ( t n T ) 2 / 2 s ( z ) i 2 π s ( z ) , for z 0 .
A n ( z ) = A n ( 0 ) e i γ 0 | A n ( 0 ) | 2 θ ( z ) / 2 π .
i n d A n d z e i ( t n T ) 2 2 s ( z ) + γ 0 e α z 2 π | s ( z ) | k l m A k A l A m * F k l m = 0 ,
i n d A n d z δ j n + γ 0 e α z 2 π | s ( z ) | k l m A k A l A m * Y k l m , j = 0 ,
Y k l m , j = lim t 1 2 t t t F k l m e i ( τ j T ) 2 / 2 s ( z ) d τ , = lim t 1 2 t e i ( k 2 + l 2 m 2 j 2 ) T 2 / 2 s ( z ) t t e i ( k + l m j ) τ T / s ( z ) d τ .
Y k l j Y k l m , j = e i [ k 2 + l 2 ( k + l j ) 2 j 2 ] T 2 / 2 s ( z ) .
i d A j d z + γ 0 e α z 2 π | s ( z ) | k l A k ( z ) A l ( z ) A k + l j * Y k l j = 0 .
d A j d z = 0 ,
U k ( z ) = e i k 2 T 2 / 2 s ( z ) ,
Y k l j = U k U l U k + l j * e i j 2 T 2 / 2 s ( z ) .
B k ( z ) = A k ( z ) U k ( z ) .
i d B j d z + j 2 T 2 β 2 + 2 s 2 ( z ) B j + γ 0 e α z 2 π | s ( z ) | k l B k B l B k + l j * = 0 .
DFT { B j ; j m } = B ˜ m = j = N / 2 N / 2 1 B j e i 2 π j m / N .
i d B ˜ m d z β 2 + T 2 2 s 2 ( z ) k = N / 2 N / 2 1 B ˜ m k x ˜ k + γ e α z 2 π | s ( z ) | | B ˜ m | 2 B ˜ m = 0 ,
x ˜ k = DFT { j 2 ; j k } .
i d B ˜ m d z + ε k m j k B ˜ k + γ | B ˜ m | 2 B ˜ m = 0 ,
B ˜ m ( z ) = B ˜ 0 sech ( m M ) e i μ ( z ) .
B ˜ m ( 0 ) = B ˜ 0 sech ( m M ) .
A n δ ( t n T ) A n 2 π T 0 e ( t n T ) 2 2 T 0 2 ,
u in ( t ) = n = N / 2 N / 2 1 A n ( 0 ) e ( t n T ) 2 2 T 0 2 2 π T 0 ,
A n ( 0 ) = B n ( 0 ) e i n 2 T 2 2 s 0 ,
B n ( 0 ) = IDFT { B ˜ m ( 0 ) ; m n } ,
P s = β 2 + T 2 4 s 0 γ 0 T 0 2 ,
B ˜ m ( 0 ) = 2 B ˜ th sech ( m M ) .
z 0 = 2 s 0 2 π M 2 T 2 | β 2 + | .
γ P T 0 2 / π > > β 2 + T 2 / | s ( z ) | ,
B ˜ m z = i γ e α z 2 π | s ( z ) | | B ˜ m | 2 B ˜ m .
B ˜ m = Y m e i θ m .
Y m = const ,
θ m ( z ) = θ m ( 0 ) + γ | Y m | 2 0 z e α x 2 π | s ( x ) | d x .
B ˜ m ( z ) = B ˜ m ( 0 ) e λ m z ,
λ m = i γ | B ˜ m | 2 , z = 1 2 π 0 z e α x | s ( x ) | d x .
z = e α | s 0 / β 2 + | 2 π [ Ei ( α | s ( z ) β 2 + | ) Ei ( α | s 0 β 2 + | ) ] .

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