Abstract

Pulse propagation in optical fiber links with randomness in the dispersion is investigated theoretically and numerically for systems with and without dispersion management. The main effects of locally correlated noise are inevitable pulse destruction and statistical broadening. However, periodic/quasi-periodic management of fiber randomness, achieved by setting the accumulated dispersion to its nominal value (pinning), essentially decreases pulse broadening. If the pinning period is short enough, we observe a statistical state that is numerically indistinguishable from the steady state in the dispersion-managed case.

© 2002 Optical Society of America

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2001

H. Sennerud, M. Karlsson, and P. A. Andrekson, “A comparison between NRZ and RZ data formats with respect to PMD-induced system degradation,” IEEE Photonics Technol. Lett. 13, 448–450 (2001).
[CrossRef]

G. Falkovich, I. Kolokolov, V. Lebedev, and S. Turitsyn, “Statistics of soliton-bearing systems with additive noise,” Phys. Rev. E 63, 025601–1–025601–4 (2001).
[CrossRef]

M. Chertkov, I. Gabitov, and J. Moeser, “Pulse confinement in optical fibers with random dispersion,” Proc. Natl. Acad. Sci. U.S.A. 98, 14208–14211 (2001).
[CrossRef] [PubMed]

M. Chertkov, I. Gabitov, I. Kolokolov, and V. Lebedev, “Shedding and interaction of solitons in imperfect medium,” JETP Lett. 74, 391–395 (2001).
[CrossRef]

Xie Chongjin, M. Karlsson, H. Sennerud, and P. A. Andrekson, “Comparison of soliton robustness with respect to polarization-mode dispersion with first-order polarization-mode dispersion-compensated linear systems,” Opt. Lett. 26, 672–674 (2001).
[CrossRef]

F. Omenetto, M. D. Moores, D. H. Reitza, and A. J. Taylor, “Adaptive control of femtosecond pulse propagation in optical fibers,” Opt. Lett. 26, 938–940 (2001).
[CrossRef]

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

2000

P. M. Lushnikov, “On the boundary of the dispersion-managed soliton existence,” Pis'ma Zh. Eksp. Teor. Fiz. 72, 163–167 (2000).

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

F. Kh. Abdullaev and B. B. Baizakov, “Disintegration of a soliton in a dispersion-managed optical communication line with random parameters,” Opt. Lett. 25, 93–95 (2000).
[CrossRef]

L. F. Mollenauer, P. V. Mamyshev, J. Gripp, M. J. Neubelt, N. Mamysheva, L. Gruner-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]

F. Kh. Abdullaev, J. C. Bronski, and G. Papanicolaou, “Soliton perturbations and the random Kepler problem,” Physica D 135, 369–386 (2000).
[CrossRef]

1999

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

1998

D. LeGuen, A. OHare, S. Del Burgo, D. Grot, F. Favre, and T. Georges, “Narrowband 640 Gbit/s soliton DWDM transmission over 1200 km of standard fibre with 100 km–21 dB amplifier spans,” Electron. Lett. 34, 2345–2346 (1998).
[CrossRef]

L. F. Mollenauer and J. Gripp, “Enhanced range for measurements of optical-time-domain-reflection-like dispersion map,” Opt. Lett. 23, 1603–1605 (1998).
[CrossRef]

1997

M. Nakazawa, K. Suzuki, H. Kubota, A. Sahara, and E. Yamada, “100Gbit/s WDM (20Gbit/s×5 channels) soliton transmission over 10000km using in-line synchronous modulation and optical filtering,” Electron. Lett. 33, 1233–1234 (1997).
[CrossRef]

1996

S. G. Evangelides, B. M. Nyman, G. T. Harvey, L. F. Mollenauer, P. V. Mamyshev, M. L. Saylors, S. K. Korotky, U. Koren, T. A. Strasser, J. J. Veselka, J. D. Evankow, A. Lucero, J. Nagel, J. Sulhoff, J. Zyskind, P. C. Corbett, M. A. Mills, and G. A. Ferguson, “Soliton WDM transmission with and without guiding filters,” IEEE Photonics Technol. Lett. 8, 1409–1411 (1996).
[CrossRef]

L. F. Mollenauer, P. V. Mamyshev, and M. J. Neubelt, “Demonstration of soliton WDM transmission at 6 and 7× 10Gbit/s, error free over transoceanic distances,” Electron. Lett. 32, 471–473 (1996).
[CrossRef]

I. Gabitov and S. K. Turitsyn, “Breathing solitons in optical fiber links,” JETP Lett. 63, 814–819 (1996).
[CrossRef]

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

L. F. Mollenauer, P. V. Mamyshev, M. J. Neubelt, “Method for facile and accurate measurement of optical fiber dispersion maps,” Opt. Lett. 21, 1724–1726 (1996).
[CrossRef] [PubMed]

1995

P. M. Lushnikov, “Dynamic criterion for collapse,” Pis'ma Zh. Eksp. Teor. Fiz. 62, 447–452 (1995).

P. M. Lushnikov, “Dynamic criterion for collapse,” JETP Lett. 62, 461–467 (1995).

R. Ohhira, A. Hasegawa, and Y. Kodama, “Methods of constructing a long-haul soliton transmission system with fibers having a distribution in dispersion,” Opt. Lett. 20, 701–703 (1995).
[CrossRef] [PubMed]

1993

J. N. Elgin, “Perturbations of optical solitons,” Phys. Rev. A 47, 4331–4341 (1993).
[CrossRef] [PubMed]

1991

1989

1988

1986

1985

J. N. Elgin, “Inverse scattering theory with stochastic initial potentials,” Phys. Lett. 110A, 441–443 (1985).
[CrossRef]

1980

1973

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

1972

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, 62–69 (1972).

V. E. Zakharov, “Collapse of Langmuir waves,” Zh. Eksp. Teor. Fiz. 62, 1745–1759 (1972).

V. E. Zakharov, “Collapse of Langmuir waves,” Sov. Phys. JETP 35, 908–914 (1972).

1971

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Zh. Eksp. Teor. Fiz. 61, 118–134 (1971).

Abdullaev, F. Kh.

F. Kh. Abdullaev, J. C. Bronski, and G. Papanicolaou, “Soliton perturbations and the random Kepler problem,” Physica D 135, 369–386 (2000).
[CrossRef]

F. Kh. Abdullaev and B. B. Baizakov, “Disintegration of a soliton in a dispersion-managed optical communication line with random parameters,” Opt. Lett. 25, 93–95 (2000).
[CrossRef]

Andrekson, P. A.

Xie Chongjin, M. Karlsson, H. Sennerud, and P. A. Andrekson, “Comparison of soliton robustness with respect to polarization-mode dispersion with first-order polarization-mode dispersion-compensated linear systems,” Opt. Lett. 26, 672–674 (2001).
[CrossRef]

H. Sennerud, M. Karlsson, and P. A. Andrekson, “A comparison between NRZ and RZ data formats with respect to PMD-induced system degradation,” IEEE Photonics Technol. Lett. 13, 448–450 (2001).
[CrossRef]

Baizakov, B. B.

Bennion, I.

N. 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]

Blow, K. J.

N. 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]

Bronski, J. C.

F. Kh. Abdullaev, J. C. Bronski, and G. Papanicolaou, “Soliton perturbations and the random Kepler problem,” Physica D 135, 369–386 (2000).
[CrossRef]

Chen, H. H.

Chertkov, M.

M. Chertkov, I. Gabitov, I. Kolokolov, and V. Lebedev, “Shedding and interaction of solitons in imperfect medium,” JETP Lett. 74, 391–395 (2001).
[CrossRef]

M. Chertkov, I. Gabitov, and J. Moeser, “Pulse confinement in optical fibers with random dispersion,” Proc. Natl. Acad. Sci. U.S.A. 98, 14208–14211 (2001).
[CrossRef] [PubMed]

Chongjin, Xie

Cohen, L. G.

Corbett, P. C.

S. G. Evangelides, B. M. Nyman, G. T. Harvey, L. F. Mollenauer, P. V. Mamyshev, M. L. Saylors, S. K. Korotky, U. Koren, T. A. Strasser, J. J. Veselka, J. D. Evankow, A. Lucero, J. Nagel, J. Sulhoff, J. Zyskind, P. C. Corbett, M. A. Mills, and G. A. Ferguson, “Soliton WDM transmission with and without guiding filters,” IEEE Photonics Technol. Lett. 8, 1409–1411 (1996).
[CrossRef]

Del Burgo, S.

D. LeGuen, A. OHare, S. Del Burgo, D. Grot, F. Favre, and T. Georges, “Narrowband 640 Gbit/s soliton DWDM transmission over 1200 km of standard fibre with 100 km–21 dB amplifier spans,” Electron. Lett. 34, 2345–2346 (1998).
[CrossRef]

Doran, N. J.

N. 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]

Elgin, J. N.

J. N. Elgin, “Perturbations of optical solitons,” Phys. Rev. A 47, 4331–4341 (1993).
[CrossRef] [PubMed]

J. N. Elgin, “Inverse scattering theory with stochastic initial potentials,” Phys. Lett. 110A, 441–443 (1985).
[CrossRef]

Evangelides, S. G.

S. G. Evangelides, B. M. Nyman, G. T. Harvey, L. F. Mollenauer, P. V. Mamyshev, M. L. Saylors, S. K. Korotky, U. Koren, T. A. Strasser, J. J. Veselka, J. D. Evankow, A. Lucero, J. Nagel, J. Sulhoff, J. Zyskind, P. C. Corbett, M. A. Mills, and G. A. Ferguson, “Soliton WDM transmission with and without guiding filters,” IEEE Photonics Technol. Lett. 8, 1409–1411 (1996).
[CrossRef]

Evankow, J. D.

S. G. Evangelides, B. M. Nyman, G. T. Harvey, L. F. Mollenauer, P. V. Mamyshev, M. L. Saylors, S. K. Korotky, U. Koren, T. A. Strasser, J. J. Veselka, J. D. Evankow, A. Lucero, J. Nagel, J. Sulhoff, J. Zyskind, P. C. Corbett, M. A. Mills, and G. A. Ferguson, “Soliton WDM transmission with and without guiding filters,” IEEE Photonics Technol. Lett. 8, 1409–1411 (1996).
[CrossRef]

Falkovich, G.

G. Falkovich, I. Kolokolov, V. Lebedev, and S. Turitsyn, “Statistics of soliton-bearing systems with additive noise,” Phys. Rev. E 63, 025601–1–025601–4 (2001).
[CrossRef]

Favre, F.

D. LeGuen, A. OHare, S. Del Burgo, D. Grot, F. Favre, and T. Georges, “Narrowband 640 Gbit/s soliton DWDM transmission over 1200 km of standard fibre with 100 km–21 dB amplifier spans,” Electron. Lett. 34, 2345–2346 (1998).
[CrossRef]

Ferguson, G. A.

S. G. Evangelides, B. M. Nyman, G. T. Harvey, L. F. Mollenauer, P. V. Mamyshev, M. L. Saylors, S. K. Korotky, U. Koren, T. A. Strasser, J. J. Veselka, J. D. Evankow, A. Lucero, J. Nagel, J. Sulhoff, J. Zyskind, P. C. Corbett, M. A. Mills, and G. A. Ferguson, “Soliton WDM transmission with and without guiding filters,” IEEE Photonics Technol. Lett. 8, 1409–1411 (1996).
[CrossRef]

Gabitov, I.

M. Chertkov, I. Gabitov, and J. Moeser, “Pulse confinement in optical fibers with random dispersion,” Proc. Natl. Acad. Sci. U.S.A. 98, 14208–14211 (2001).
[CrossRef] [PubMed]

M. Chertkov, I. Gabitov, I. Kolokolov, and V. Lebedev, “Shedding and interaction of solitons in imperfect medium,” JETP Lett. 74, 391–395 (2001).
[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]

I. Gabitov and S. K. Turitsyn, “Breathing solitons in optical fiber links,” JETP Lett. 63, 814–819 (1996).
[CrossRef]

Georges, T.

D. LeGuen, A. OHare, S. Del Burgo, D. Grot, F. Favre, and T. Georges, “Narrowband 640 Gbit/s soliton DWDM transmission over 1200 km of standard fibre with 100 km–21 dB amplifier spans,” Electron. Lett. 34, 2345–2346 (1998).
[CrossRef]

Gisin, N.

N. Gisin, “Solutions of the dynamical equation for polarization dispersion,” Opt. Commun. 86, 371–373 (1991).
[CrossRef]

Gordon, J. P.

Gripp, J.

Grot, D.

D. LeGuen, A. OHare, S. Del Burgo, D. Grot, F. Favre, and T. Georges, “Narrowband 640 Gbit/s soliton DWDM transmission over 1200 km of standard fibre with 100 km–21 dB amplifier spans,” Electron. Lett. 34, 2345–2346 (1998).
[CrossRef]

Gruner-Nielsen, L.

Harvey, G. T.

S. G. Evangelides, B. M. Nyman, G. T. Harvey, L. F. Mollenauer, P. V. Mamyshev, M. L. Saylors, S. K. Korotky, U. Koren, T. A. Strasser, J. J. Veselka, J. D. Evankow, A. Lucero, J. Nagel, J. Sulhoff, J. Zyskind, P. C. Corbett, M. A. Mills, and G. A. Ferguson, “Soliton WDM transmission with and without guiding filters,” IEEE Photonics Technol. Lett. 8, 1409–1411 (1996).
[CrossRef]

Hasegawa, A.

R. Ohhira, A. Hasegawa, and Y. Kodama, “Methods of constructing a long-haul soliton transmission system with fibers having a distribution in dispersion,” Opt. Lett. 20, 701–703 (1995).
[CrossRef] [PubMed]

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

Haus, H. A.

Karlsson, M.

Xie Chongjin, M. Karlsson, H. Sennerud, and P. A. Andrekson, “Comparison of soliton robustness with respect to polarization-mode dispersion with first-order polarization-mode dispersion-compensated linear systems,” Opt. Lett. 26, 672–674 (2001).
[CrossRef]

H. Sennerud, M. Karlsson, and P. A. Andrekson, “A comparison between NRZ and RZ data formats with respect to PMD-induced system degradation,” IEEE Photonics Technol. Lett. 13, 448–450 (2001).
[CrossRef]

Knox, F. M.

N. 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]

Kodama, Y.

Kogelnik, H.

Kolokolov, I.

G. Falkovich, I. Kolokolov, V. Lebedev, and S. Turitsyn, “Statistics of soliton-bearing systems with additive noise,” Phys. Rev. E 63, 025601–1–025601–4 (2001).
[CrossRef]

M. Chertkov, I. Gabitov, I. Kolokolov, and V. Lebedev, “Shedding and interaction of solitons in imperfect medium,” JETP Lett. 74, 391–395 (2001).
[CrossRef]

Koren, U.

S. G. Evangelides, B. M. Nyman, G. T. Harvey, L. F. Mollenauer, P. V. Mamyshev, M. L. Saylors, S. K. Korotky, U. Koren, T. A. Strasser, J. J. Veselka, J. D. Evankow, A. Lucero, J. Nagel, J. Sulhoff, J. Zyskind, P. C. Corbett, M. A. Mills, and G. A. Ferguson, “Soliton WDM transmission with and without guiding filters,” IEEE Photonics Technol. Lett. 8, 1409–1411 (1996).
[CrossRef]

Korotky, S. K.

S. G. Evangelides, B. M. Nyman, G. T. Harvey, L. F. Mollenauer, P. V. Mamyshev, M. L. Saylors, S. K. Korotky, U. Koren, T. A. Strasser, J. J. Veselka, J. D. Evankow, A. Lucero, J. Nagel, J. Sulhoff, J. Zyskind, P. C. Corbett, M. A. Mills, and G. A. Ferguson, “Soliton WDM transmission with and without guiding filters,” IEEE Photonics Technol. Lett. 8, 1409–1411 (1996).
[CrossRef]

Kubota, H.

M. Nakazawa, K. Suzuki, H. Kubota, A. Sahara, and E. Yamada, “100Gbit/s WDM (20Gbit/s×5 channels) soliton transmission over 10000km using in-line synchronous modulation and optical filtering,” Electron. Lett. 33, 1233–1234 (1997).
[CrossRef]

Lebedev, V.

M. Chertkov, I. Gabitov, I. Kolokolov, and V. Lebedev, “Shedding and interaction of solitons in imperfect medium,” JETP Lett. 74, 391–395 (2001).
[CrossRef]

G. Falkovich, I. Kolokolov, V. Lebedev, and S. Turitsyn, “Statistics of soliton-bearing systems with additive noise,” Phys. Rev. E 63, 025601–1–025601–4 (2001).
[CrossRef]

LeGuen, D.

D. LeGuen, A. OHare, S. Del Burgo, D. Grot, F. Favre, and T. Georges, “Narrowband 640 Gbit/s soliton DWDM transmission over 1200 km of standard fibre with 100 km–21 dB amplifier spans,” Electron. Lett. 34, 2345–2346 (1998).
[CrossRef]

Lin, C.

Lucero, A.

S. G. Evangelides, B. M. Nyman, G. T. Harvey, L. F. Mollenauer, P. V. Mamyshev, M. L. Saylors, S. K. Korotky, U. Koren, T. A. Strasser, J. J. Veselka, J. D. Evankow, A. Lucero, J. Nagel, J. Sulhoff, J. Zyskind, P. C. Corbett, M. A. Mills, and G. A. Ferguson, “Soliton WDM transmission with and without guiding filters,” IEEE Photonics Technol. Lett. 8, 1409–1411 (1996).
[CrossRef]

Lushnikov, P. M.

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

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

P. M. Lushnikov, “On the boundary of the dispersion-managed soliton existence,” Pis'ma Zh. Eksp. Teor. Fiz. 72, 163–167 (2000).

P. M. Lushnikov, “Dynamic criterion for collapse,” JETP Lett. 62, 461–467 (1995).

P. M. Lushnikov, “Dynamic criterion for collapse,” Pis'ma Zh. Eksp. Teor. Fiz. 62, 447–452 (1995).

Mamyshev, P. V.

L. F. Mollenauer, P. V. Mamyshev, J. Gripp, M. J. Neubelt, N. Mamysheva, L. Gruner-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]

L. F. Mollenauer, P. V. Mamyshev, and M. J. Neubelt, “Demonstration of soliton WDM transmission at 6 and 7× 10Gbit/s, error free over transoceanic distances,” Electron. Lett. 32, 471–473 (1996).
[CrossRef]

L. F. Mollenauer, P. V. Mamyshev, M. J. Neubelt, “Method for facile and accurate measurement of optical fiber dispersion maps,” Opt. Lett. 21, 1724–1726 (1996).
[CrossRef] [PubMed]

S. G. Evangelides, B. M. Nyman, G. T. Harvey, L. F. Mollenauer, P. V. Mamyshev, M. L. Saylors, S. K. Korotky, U. Koren, T. A. Strasser, J. J. Veselka, J. D. Evankow, A. Lucero, J. Nagel, J. Sulhoff, J. Zyskind, P. C. Corbett, M. A. Mills, and G. A. Ferguson, “Soliton WDM transmission with and without guiding filters,” IEEE Photonics Technol. Lett. 8, 1409–1411 (1996).
[CrossRef]

Mamysheva, N.

Menyuk, C. R.

Mezentsev, V. K.

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

Mills, M. A.

S. G. Evangelides, B. M. Nyman, G. T. Harvey, L. F. Mollenauer, P. V. Mamyshev, M. L. Saylors, S. K. Korotky, U. Koren, T. A. Strasser, J. J. Veselka, J. D. Evankow, A. Lucero, J. Nagel, J. Sulhoff, J. Zyskind, P. C. Corbett, M. A. Mills, and G. A. Ferguson, “Soliton WDM transmission with and without guiding filters,” IEEE Photonics Technol. Lett. 8, 1409–1411 (1996).
[CrossRef]

Moeser, J.

M. Chertkov, I. Gabitov, and J. Moeser, “Pulse confinement in optical fibers with random dispersion,” Proc. Natl. Acad. Sci. U.S.A. 98, 14208–14211 (2001).
[CrossRef] [PubMed]

Mollenauer, L. F.

Moores, M. D.

Nagel, J.

S. G. Evangelides, B. M. Nyman, G. T. Harvey, L. F. Mollenauer, P. V. Mamyshev, M. L. Saylors, S. K. Korotky, U. Koren, T. A. Strasser, J. J. Veselka, J. D. Evankow, A. Lucero, J. Nagel, J. Sulhoff, J. Zyskind, P. C. Corbett, M. A. Mills, and G. A. Ferguson, “Soliton WDM transmission with and without guiding filters,” IEEE Photonics Technol. Lett. 8, 1409–1411 (1996).
[CrossRef]

Nagel, J. A.

Nakazawa, M.

M. Nakazawa, K. Suzuki, H. Kubota, A. Sahara, and E. Yamada, “100Gbit/s WDM (20Gbit/s×5 channels) soliton transmission over 10000km using in-line synchronous modulation and optical filtering,” Electron. Lett. 33, 1233–1234 (1997).
[CrossRef]

Neubelt, M. J.

Nyman, B. M.

S. G. Evangelides, B. M. Nyman, G. T. Harvey, L. F. Mollenauer, P. V. Mamyshev, M. L. Saylors, S. K. Korotky, U. Koren, T. A. Strasser, J. J. Veselka, J. D. Evankow, A. Lucero, J. Nagel, J. Sulhoff, J. Zyskind, P. C. Corbett, M. A. Mills, and G. A. Ferguson, “Soliton WDM transmission with and without guiding filters,” IEEE Photonics Technol. Lett. 8, 1409–1411 (1996).
[CrossRef]

OHare, A.

D. LeGuen, A. OHare, S. Del Burgo, D. Grot, F. Favre, and T. Georges, “Narrowband 640 Gbit/s soliton DWDM transmission over 1200 km of standard fibre with 100 km–21 dB amplifier spans,” Electron. Lett. 34, 2345–2346 (1998).
[CrossRef]

Ohhira, R.

Omenetto, F.

Papanicolaou, G.

F. Kh. Abdullaev, J. C. Bronski, and G. Papanicolaou, “Soliton perturbations and the random Kepler problem,” Physica D 135, 369–386 (2000).
[CrossRef]

Poole, C. D.

Reitza, D. H.

Sahara, A.

M. Nakazawa, K. Suzuki, H. Kubota, A. Sahara, and E. Yamada, “100Gbit/s WDM (20Gbit/s×5 channels) soliton transmission over 10000km using in-line synchronous modulation and optical filtering,” Electron. Lett. 33, 1233–1234 (1997).
[CrossRef]

Saylors, M. L.

S. G. Evangelides, B. M. Nyman, G. T. Harvey, L. F. Mollenauer, P. V. Mamyshev, M. L. Saylors, S. K. Korotky, U. Koren, T. A. Strasser, J. J. Veselka, J. D. Evankow, A. Lucero, J. Nagel, J. Sulhoff, J. Zyskind, P. C. Corbett, M. A. Mills, and G. A. Ferguson, “Soliton WDM transmission with and without guiding filters,” IEEE Photonics Technol. Lett. 8, 1409–1411 (1996).
[CrossRef]

Schafer, T.

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

Sennerud, H.

Xie Chongjin, M. Karlsson, H. Sennerud, and P. A. Andrekson, “Comparison of soliton robustness with respect to polarization-mode dispersion with first-order polarization-mode dispersion-compensated linear systems,” Opt. Lett. 26, 672–674 (2001).
[CrossRef]

H. Sennerud, M. Karlsson, and P. A. Andrekson, “A comparison between NRZ and RZ data formats with respect to PMD-induced system degradation,” IEEE Photonics Technol. Lett. 13, 448–450 (2001).
[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, 62–69 (1972).

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Zh. Eksp. Teor. Fiz. 61, 118–134 (1971).

Smith, K.

Smith, N.

N. 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. Schafer, K. H. Spatschek, and V. K. Mezentsev, “Path-averaged chirped optical soliton in dispersion-managed fiber communication lines,” Opt. Commun. 163, 122–158 (1999).
[CrossRef]

Strasser, T. A.

S. G. Evangelides, B. M. Nyman, G. T. Harvey, L. F. Mollenauer, P. V. Mamyshev, M. L. Saylors, S. K. Korotky, U. Koren, T. A. Strasser, J. J. Veselka, J. D. Evankow, A. Lucero, J. Nagel, J. Sulhoff, J. Zyskind, P. C. Corbett, M. A. Mills, and G. A. Ferguson, “Soliton WDM transmission with and without guiding filters,” IEEE Photonics Technol. Lett. 8, 1409–1411 (1996).
[CrossRef]

Sulhoff, J.

S. G. Evangelides, B. M. Nyman, G. T. Harvey, L. F. Mollenauer, P. V. Mamyshev, M. L. Saylors, S. K. Korotky, U. Koren, T. A. Strasser, J. J. Veselka, J. D. Evankow, A. Lucero, J. Nagel, J. Sulhoff, J. Zyskind, P. C. Corbett, M. A. Mills, and G. A. Ferguson, “Soliton WDM transmission with and without guiding filters,” IEEE Photonics Technol. Lett. 8, 1409–1411 (1996).
[CrossRef]

Suzuki, K.

M. Nakazawa, K. Suzuki, H. Kubota, A. Sahara, and E. Yamada, “100Gbit/s WDM (20Gbit/s×5 channels) soliton transmission over 10000km using in-line synchronous modulation and optical filtering,” Electron. Lett. 33, 1233–1234 (1997).
[CrossRef]

Tappert, F.

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

Taylor, A. J.

Turitsyn, S.

G. Falkovich, I. Kolokolov, V. Lebedev, and S. Turitsyn, “Statistics of soliton-bearing systems with additive noise,” Phys. Rev. E 63, 025601–1–025601–4 (2001).
[CrossRef]

Turitsyn, S. K.

S. K. Turitsyn, T. Schafer, K. H. Spatschek, and V. K. Mezentsev, “Path-averaged chirped optical soliton in dispersion-managed fiber communication lines,” Opt. Commun. 163, 122–158 (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]

I. Gabitov and S. K. Turitsyn, “Breathing solitons in optical fiber links,” JETP Lett. 63, 814–819 (1996).
[CrossRef]

Veng, T.

Veselka, J. J.

S. G. Evangelides, B. M. Nyman, G. T. Harvey, L. F. Mollenauer, P. V. Mamyshev, M. L. Saylors, S. K. Korotky, U. Koren, T. A. Strasser, J. J. Veselka, J. D. Evankow, A. Lucero, J. Nagel, J. Sulhoff, J. Zyskind, P. C. Corbett, M. A. Mills, and G. A. Ferguson, “Soliton WDM transmission with and without guiding filters,” IEEE Photonics Technol. Lett. 8, 1409–1411 (1996).
[CrossRef]

Wai, P. K.

Winters, J. H.

Yamada, E.

M. Nakazawa, K. Suzuki, H. Kubota, A. Sahara, and E. Yamada, “100Gbit/s WDM (20Gbit/s×5 channels) soliton transmission over 10000km using in-line synchronous modulation and optical filtering,” Electron. Lett. 33, 1233–1234 (1997).
[CrossRef]

Zakharov, V. E.

V. E. Zakharov, “Collapse of Langmuir waves,” Zh. Eksp. Teor. Fiz. 62, 1745–1759 (1972).

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, 62–69 (1972).

V. E. Zakharov, “Collapse of Langmuir waves,” Sov. Phys. JETP 35, 908–914 (1972).

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Zh. Eksp. Teor. Fiz. 61, 118–134 (1971).

Zyskind, J.

S. G. Evangelides, B. M. Nyman, G. T. Harvey, L. F. Mollenauer, P. V. Mamyshev, M. L. Saylors, S. K. Korotky, U. Koren, T. A. Strasser, J. J. Veselka, J. D. Evankow, A. Lucero, J. Nagel, J. Sulhoff, J. Zyskind, P. C. Corbett, M. A. Mills, and G. A. Ferguson, “Soliton WDM transmission with and without guiding filters,” IEEE Photonics Technol. Lett. 8, 1409–1411 (1996).
[CrossRef]

Appl. Phys. Lett.

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

Electron. Lett.

L. F. Mollenauer, P. V. Mamyshev, and M. J. Neubelt, “Demonstration of soliton WDM transmission at 6 and 7× 10Gbit/s, error free over transoceanic distances,” Electron. Lett. 32, 471–473 (1996).
[CrossRef]

M. Nakazawa, K. Suzuki, H. Kubota, A. Sahara, and E. Yamada, “100Gbit/s WDM (20Gbit/s×5 channels) soliton transmission over 10000km using in-line synchronous modulation and optical filtering,” Electron. Lett. 33, 1233–1234 (1997).
[CrossRef]

D. LeGuen, A. OHare, S. Del Burgo, D. Grot, F. Favre, and T. Georges, “Narrowband 640 Gbit/s soliton DWDM transmission over 1200 km of standard fibre with 100 km–21 dB amplifier spans,” Electron. Lett. 34, 2345–2346 (1998).
[CrossRef]

N. 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]

IEEE Photonics Technol. Lett.

H. Sennerud, M. Karlsson, and P. A. Andrekson, “A comparison between NRZ and RZ data formats with respect to PMD-induced system degradation,” IEEE Photonics Technol. Lett. 13, 448–450 (2001).
[CrossRef]

S. G. Evangelides, B. M. Nyman, G. T. Harvey, L. F. Mollenauer, P. V. Mamyshev, M. L. Saylors, S. K. Korotky, U. Koren, T. A. Strasser, J. J. Veselka, J. D. Evankow, A. Lucero, J. Nagel, J. Sulhoff, J. Zyskind, P. C. Corbett, M. A. Mills, and G. A. Ferguson, “Soliton WDM transmission with and without guiding filters,” IEEE Photonics Technol. Lett. 8, 1409–1411 (1996).
[CrossRef]

JETP Lett.

I. Gabitov and S. K. Turitsyn, “Breathing solitons in optical fiber links,” JETP Lett. 63, 814–819 (1996).
[CrossRef]

M. Chertkov, I. Gabitov, I. Kolokolov, and V. Lebedev, “Shedding and interaction of solitons in imperfect medium,” JETP Lett. 74, 391–395 (2001).
[CrossRef]

P. M. Lushnikov, “Dynamic criterion for collapse,” JETP Lett. 62, 461–467 (1995).

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

Opt. Commun.

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

N. Gisin, “Solutions of the dynamical equation for polarization dispersion,” Opt. Commun. 86, 371–373 (1991).
[CrossRef]

Opt. Lett.

F. Kh. Abdullaev and B. B. Baizakov, “Disintegration of a soliton in a dispersion-managed optical communication line with random parameters,” Opt. Lett. 25, 93–95 (2000).
[CrossRef]

L. F. Mollenauer, P. V. Mamyshev, J. Gripp, M. J. Neubelt, N. Mamysheva, L. Gruner-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]

J. P. Gordon and H. A. Haus, “Random walk of coherently amplified solitons in optical fiber transmission,” Opt. Lett. 11, 665–667 (1986).
[CrossRef] [PubMed]

C. D. Poole, “Statistical treatment of polarization dispersion in single-mode fiber,” Opt. Lett. 13, 687–689 (1988).
[CrossRef] [PubMed]

L. F. Mollenauer, K. Smith, J. P. Gordon, and C. R. Menyuk, “Resistance of solitons to the effects of polarization dispersion in optical fibers,” Opt. Lett. 14, 1219–1221 (1989).
[CrossRef] [PubMed]

C. D. Poole, J. H. Winters, and J. A. Nagel, “Dynamical equation for polarization dispersion,” Opt. Lett. 16, 372–374 (1991).
[CrossRef] [PubMed]

P. K. Wai, C. R. Menyuk, and H. H. Chen, “Stability of solitons in randomly varying birefringent fibers,” Opt. Lett. 16, 1231–1233 (1991).
[CrossRef] [PubMed]

R. Ohhira, A. Hasegawa, and Y. Kodama, “Methods of constructing a long-haul soliton transmission system with fibers having a distribution in dispersion,” Opt. Lett. 20, 701–703 (1995).
[CrossRef] [PubMed]

L. F. Mollenauer and J. Gripp, “Enhanced range for measurements of optical-time-domain-reflection-like dispersion map,” Opt. Lett. 23, 1603–1605 (1998).
[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]

L. F. Mollenauer, P. V. Mamyshev, M. J. Neubelt, “Method for facile and accurate measurement of optical fiber dispersion maps,” Opt. Lett. 21, 1724–1726 (1996).
[CrossRef] [PubMed]

Xie Chongjin, M. Karlsson, H. Sennerud, and P. A. Andrekson, “Comparison of soliton robustness with respect to polarization-mode dispersion with first-order polarization-mode dispersion-compensated linear systems,” Opt. Lett. 26, 672–674 (2001).
[CrossRef]

F. Omenetto, M. D. Moores, D. H. Reitza, and A. J. Taylor, “Adaptive control of femtosecond pulse propagation in optical fibers,” Opt. Lett. 26, 938–940 (2001).
[CrossRef]

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

Phys. Lett.

J. N. Elgin, “Inverse scattering theory with stochastic initial potentials,” Phys. Lett. 110A, 441–443 (1985).
[CrossRef]

Phys. Rev. A

J. N. Elgin, “Perturbations of optical solitons,” Phys. Rev. A 47, 4331–4341 (1993).
[CrossRef] [PubMed]

Phys. Rev. E

G. Falkovich, I. Kolokolov, V. Lebedev, and S. Turitsyn, “Statistics of soliton-bearing systems with additive noise,” Phys. Rev. E 63, 025601–1–025601–4 (2001).
[CrossRef]

Physica D

F. Kh. Abdullaev, J. C. Bronski, and G. Papanicolaou, “Soliton perturbations and the random Kepler problem,” Physica D 135, 369–386 (2000).
[CrossRef]

Pis'ma Zh. Eksp. Teor. Fiz.

P. M. Lushnikov, “Dynamic criterion for collapse,” Pis'ma Zh. Eksp. Teor. Fiz. 62, 447–452 (1995).

P. M. Lushnikov, “On the boundary of the dispersion-managed soliton existence,” Pis'ma Zh. Eksp. Teor. Fiz. 72, 163–167 (2000).

Proc. Natl. Acad. Sci. U.S.A.

M. Chertkov, I. Gabitov, and J. Moeser, “Pulse confinement in optical fibers with random dispersion,” Proc. Natl. Acad. Sci. U.S.A. 98, 14208–14211 (2001).
[CrossRef] [PubMed]

Sov. Phys. JETP

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, 62–69 (1972).

V. E. Zakharov, “Collapse of Langmuir waves,” Sov. Phys. JETP 35, 908–914 (1972).

Zh. Eksp. Teor. Fiz.

V. E. Zakharov, “Collapse of Langmuir waves,” Zh. Eksp. Teor. Fiz. 62, 1745–1759 (1972).

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Zh. Eksp. Teor. Fiz. 61, 118–134 (1971).

Other

G. P. Agrawal, Fiber-Optic Communication Systems (Wiley, New York, 1997), Chap. 4, pp. 170–173.

D. LeGuen, S. Del Burgo, L. Moulinard, D. Grot, M. Henry, F. Favre, and T. Georges, “Narrow band 1.02 Tbit/s (51*20 Gbit/s) soliton DWDM transmission over 1000 km of standard fiber with 100 km amplifier spans,” in Optical Fiber Communication Conference, Postconference Digest, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), paper PD4–1.

L. F. Mollenauer, “Dispersion maps for ultra long distance, terabit capacity WDM,” Conference on Lasers and Electro-Optics, Postconference Digest, Vol. 39 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2000), p. 137.

Y. Kodama, “Nonlinear chirped RZ and NRZ pulses in optical transmission lines,” in Proceedings of the International Symposium on New Trends in Optical Soliton Transmission Systems, A. Hasegawa, ed. (Kluwer Academic, Boston, Mass., 1998), pp. 131–153.

N. J. Doran, W. Forysiak, and J. H. B. Nijhof, “Remarkable features of DM solitons: implications for high speed and WDM systems,” in Proceedings of the International Symposium on New Trends in Optical Soliton Transmission Systems, A. Hasegawa, ed. (Kluwer Academic, Boston, Mass., 1998), pp. 303–316.

W. Feller, An Introduction to Probability Theory and Its Applications (Wiley, New York, 1957).

F. Kh. Abdullaev, J. G. Caputo, and M. P. Srensen, “Propagation and interaction of optical solitons in fibers with random parameters,” in Proceedings of the International Symposium on New Trends in Optical Soliton Transmission Systems, A. Hasegawa, ed. (Kluwer Academic, Boston, Mass., 1998), pp. 1–13.

B. D. Hughes, Random Walks and Random Environments. Vol. 1. Random Walks (Clarendon, Oxford, 1995).

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Figures (4)

Fig. 1
Fig. 1

Kernels and stationary solution shapes for the path-averaged equation: constant-dispersion case (model A). The upper row shows the shape of the kernel (16) for various pinning lengths. The lower row shows the corresponding stationary solutions (log scale) for the path-averaged equation. The pictures in the left column are for noise strength D=0.1, and those on the right are for D=2.5.

Fig. 2
Fig. 2

Kernels and stationary solution shapes for the path-averaged equation: dispersion-management case (model B). The upper row shows the shape of the kernel for various pinning lengths. The lower row shows the corresponding stationary solutions (log scale) for the path-averaged equation. The pictures in the left column are for noise strength D=0.1, and those on the right are for D=2.5.

Fig. 3
Fig. 3

Statistical description of the effect of pinning on optical pulse dynamics: constant-dispersion case (model A). The upper-left subfigure is a comparison of a pinned and unpinned random-dispersion profile. The upper center and right subfigures are PDFs for pulse amplitude and width, measured at distance z=95 for noise strength D=0.1, for various pinning lengths. The left and center subfigures in the middle row show the dependence of the average amplitude and width on the distance z for various pinning lengths and noise strength D=0.1. The right subfigure of the middle row and the left subfigure of the bottom row represent PDFs of the pulse amplitude and width measured at distance z=95 for noise strength D=2.5 for various pinning lengths. The center and right subfigures of the bottom row show the dependence of the average amplitude and width on the distance z for various pinning lengths and noise strength D=2.5.

Fig. 4
Fig. 4

Statistical description of the effect of pinning on the optical pulse dynamics: dispersion-management case (model B). The upper-left subfigure is a comparison of pinned and unpinned dispersion maps with randomness. The upper center and right subfigures are PDFs for pulse amplitude and width, measured at distance z=95 for noise strength D=0.1, for various pinning lengths. The left and center subfigures in the middle row show the dependence of the average amplitude and width on the distance z for various pinning lengths and noise strength D=0.1. The right subfigure of the middle row and the left subfigure of the bottom row represent PDFs of the pulse amplitude and width measured at distance z=95 for noise strength D=2.5 for various pinning lengths. The center and right subfigures of the bottom row show the dependence of the average amplitude and width on the distance z for various pinning lengths and noise strength D=2.5.

Equations (51)

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-izψ=d(z)t2ψ+2ψ2ψ¯.
d(z)=ddet(z)+ξ(z),
ξ(z1)ξ(z2)=Dδ(z1-z2).
ξp(y)ξp(z)=Dδ(z-y)-1lj+1-lj,
ξp(z)=ξ(z)-1li+1-lilili+1dyξ(y),
ξpp(z)=ξ(z)-jδ(z-lj)lj-1ljdzξ(z).
ψ(z; t)=-dω exp-iωt+ω20z[ξ(z)+ddet(z)-d0]dzϕω(z).
-izϕω+d0ω2ϕω
=2-δ(ω1+ω2-ω3-ω)dω1dω2dω3ϕ1ϕ2ϕ¯3
×exp-iΔ0z[ξ(z)+ddet(z)-d0]dz,
Δ=ω12+ω22-ω32-ω2,ϕiϕωi,i=1, 2, 3.
F(Δ; z)
z-10zdz1 exp-iΔ0z1[ξ(z)+ddet(z)-d0]dz.
-izφω+d0ω2φω=2-δ(ω1+ω2-ω3-ω)×dω1dω2dω3F(Δ; z)φ1φ2φ¯3.
F(Δ; z)=2Δ2Dz1-exp-Δ2Dz12,
F(Δ; z)2=1Δ4D2z21-43 exp-Δ2Dz2+13 exp(-2Δ2Dz),
zξ=b4/D,
F(Δ; z)exp-Δ2Dz(l-z)2+i0zdz[ddet(z)-d0]zl,
(formodelA)=exp-Δ2Dl4πΔ2Dl  ErfiΔ2Dl4.
R(ω)=1π dsFˆ(s)exp(-isω2)δ(ω1+ω2-ω3-ω)dω1dω2dω3φ1(s)φ2(s)φ¯3(s),
Fˆ-1(R(ω))=2dsGˆ(s)(Ψ(s)(t)),
Az=(d0+ξ)B,B2it(ψψt*-ψ*ψt)dt.
Bz=8[(d0+ξ(z))]X-4Y,
2d0X+2ξX-Y=0.
2d0X=Y,
Az=[ξ(z)+d0]B(0)+8[ξ(z)+d0]0z[ξ(z)+d0]X(z)dz-4[ξ(z)+d0]0zY(z)dz.
Az=d0B(0)+8d020zX(z)dz+4DX(z)-4d00zY(z)dz.
β(TRMS(z)2-TRMS(0)2)/TRMS(0)2dvar2zvarz/b4.
-izψ=d(z)t2ψ.
ψ(t; z)=-+dk-+ dt2π expik(t-t)-ik20zd(z)dzψ(t; 0).
ψ(t; z)=a0 exp(iλ0)×exp-t2 1-ib02μ0b02+4(i+b02μ0)0zd(z)dz×1+4(i/b02+μ)0zd(z)dz-1.
a(z)=a0  abs1+4(i/b02+μ)0zd(z)dz-1/2,
λ(z)=λ0+arg1+4(i/b02+μ)0zd(z)dz-1/2,
b(z)=b021+4µ00zd(z)dz2+16b020zd(z)dz2,
μ(z)=b04μ0+4(1+b04μ02)0zd(z)dzb041+4µ00zd(z)dz2+160zd(z)dz2.
P(al|l; a0, b0, μ0)=2a04(al10πDl{μ02+[(a0/al)4-1](μ02+1/b04)})-1/2×k=1;2 exp-(μ0{1+(-1)k1+[(a0/al)4-1][1+1/(μ02b04)]}[4(μ02+1/b04)]-1+d0l)22Dl.
P(bl|l; a0, b0, μ0)=bl(32πDl[μ02b04+(bl2/b02-1)(1+b04μ02)])-1/2×k=1;2 exp-(μ0b04[1+(-1)k1+(bl2/b02-1)(1+1/b04μ02)][4(1+b04μ02)]-1+d0l)22Dl,
P(μl|l; a0, b0, μ0)=k=1;21+b04μ02(1+b04μ02)2-4b04μl2+(-1)k(128μl4πDl)-1×exp-12Dl1+b04μ0(μ0-2µl)-(-1)k(1+b04μ02)2-4b04μl28µl(1+b04μ02)-d0l2.
p(h, z)=14πDz exp-h24Dz.
p(h, z)=0zdzf(h, z)p(0, z-z).
gˆ(u)=0dz exp(-uz)g(z),Re(u)>0.
pˆ(x, u)=fˆ(x, u)pˆ(0, u).
pˆ(x, u)=14Du exp-ux2D,
fˆ(x, u)=exp-ux2D.
f(h, z)=h24πDz3 exp-h24Dz,t>0.
f(h, z)|h|2πDz-3/2.
0zdzzf(h, z)=2π h24D Z1/2 exp-h24DZ-h22D1-Φh24D1/4,
τn(s)=j=0njFj(s)Rn(s).
F(s; ξ)=1-1-ξ2,s=01-1-ξ2ξ|s|,s0.
τ(s)=j=1jFj(s)=ξF(s; ξ)ξ1-,
τn(s)=2π  n,s=02π|s|2π  n,s0.

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