G. Genty, J. M. Dudley, and B. J. Eggleton, “Modulation control and spectral shaping of optical fiber supercontinuum generation in the picosecond regime,” Appl. Phys. B 94, 187–194 (2009).

[CrossRef]

A. Hause and F. Mitschke, “Reduced soliton interaction by Raman self-frequency-shift,” Phys. Rev. A 80, 063824 (2009).

[CrossRef]

A. Peleg, “Energy exchange in fast optical soliton collisions as a random cascade model,” Phys. Lett. A 373, 2734–2738 (2009).

[CrossRef]

Y. Chung and A. Peleg, “Monte Carlo simulations of pulse propagation in massive multichannel optical fiber communication systems,” Phys. Rev. A 77, 063835 (2008).

[CrossRef]

N. Korneev, E. A. Kuzin, B. Ibarra-Escamilla, M. Bello-Jiménez, and A. Flores-Rosas, “Initial development of supercontinuum in fibers with anomalous dispersion pumped by nanosecond-long pulses,” Opt. Express 16, 2636–2645 (2008).

[CrossRef]
[PubMed]

A. Peleg, “Intermittent dynamics, strong correlations, and bit-error-rate in multichannel optical fiber communication systems,” Phys. Lett. A 360, 533–538 (2007).

[CrossRef]

F. Luan, D. V. Skryabin, A. V. Yulin, and J. C. Knight, “Energy exchange between colliding solitons in photonic crystal fibers,” Opt. Express 14, 9844–9853 (2006).

[CrossRef]
[PubMed]

M. H. Frosz, O. Bang, and A. Bjarklev, “Soliton collision and Raman gain regimes in continuous-wave pumped supercontinuum generation,” Opt. Express 14, 9391–9407 (2006).

[CrossRef]
[PubMed]

M. Olivier, V. Roy, and M. Pichè, “Influence of the Raman effect on bound states of dissipative solitons,” Opt. Express 14, 9728–9742 (2006).

[CrossRef]
[PubMed]

Y. Chung and A. Peleg, “Strongly non-Gaussian statistics of optical soliton parameters due to collisions in the presence of delayed Raman response,” Nonlinearity 18, 1555–1574 (2005).

[CrossRef]

A. Peleg, “Log-normal distribution of pulse amplitudes due to Raman cross talk in wavelength division multiplexing soliton transmission,” Opt. Lett. 29, 1980–1982 (2004).

[CrossRef]
[PubMed]

A. Peleg, M. Chertkov, and I. Gabitov, “Inelastic interchannel collisions of pulses in optical fibers in the presence of third-order dispersion,” J. Opt. Soc. Am. B 21, 18–23 (2004).

[CrossRef]

J. Soneson and A. Peleg, “Effect of quintic nonlinearity on soliton collisions in optical fibers,” Physica D 195, 123–140 (2004).

[CrossRef]

A. Efimov, A. J. Taylor, F. G. Omenetto, and E. Vanin, “Adaptive control of femtosecond soliton self-frequency shift in fibers,” Opt. Lett. 29, 271–273 (2004).

[CrossRef]
[PubMed]

M. Muktoyuk and S. Kumar, “Noise variance due to stimulated Raman scattering among channels of a wavelength-division-multiplexed system,” IEEE Photon. Technol. Lett. 15, 1222–1224 (2003).

[CrossRef]

T. Yamamoto and S. Norimatsu, “Statistical analysis on stimulated Raman crosstalk in dispersion-managed fiber links,” J. Lightwave Technol. 21, 2229–2239 (2003).

[CrossRef]

A. Peleg and Y. Chung, “Stationary solutions to the nonlinear Schrödinger equation in the presence of third order dispersion,” J. Phys. A 36, 10039–10051 (2003).

[CrossRef]

M. Chertkov, Y. Chung, A. Dyachenko, I. Gabitov, I. Kolokolov, and V. Lebedev, “Shedding and interaction of solitons in weakly disordered optical fibers,” Phys. Rev. E 67, 036615 (2003).

[CrossRef]

A. Peleg, M. Chertkov, and I. Gabitov, “Interchannel interaction of optical solitons,” Phys. Rev. E 68, 026605 (2003).

[CrossRef]

E. A. Kuznetsov, A. V. Mikhailov, and I. A. Shimokhin, “Nonlinear interaction of solitons and radiation,” Physica D 87, 201–215 (1995).

[CrossRef]

F. Forghieri, R. W. Tkach, and A. R. Chraplyvy, “Effect of modulation statistics on Raman crosstalk in WDM systems,” IEEE Photon. Technol. Lett. 7, 101–103 (1995).

[CrossRef]

B. A. Malomed, “Soliton-collision problem in the nonlinear Schrödinger equation with a nonlinear damping term,” Phys. Rev. A 44, 1412–1414 (1991).

[CrossRef]
[PubMed]

D. J. Kaup, “Second-order perturbations for solitons in optical fibers,” Phys. Rev. A 44, 4582–4590 (1991).

[CrossRef]
[PubMed]

B. A. Malomed, “Radiative losses in soliton-soliton collisions in an optical fiber with the third-order dispersion,” Phys. Rev. A 43, 3114–3116 (1991).

[CrossRef]
[PubMed]

H. Yoshida, “Construction of higher order symplectic integrators,” Phys. Lett. A 150, 262–268 (1990).

[CrossRef]

Y. Kodama and A. Hasegawa, “Nonlinear pulse propagation in a monomode dielectric guide,” IEEE J. Quantum Electron. 23, 510–524 (1987).

[CrossRef]

A. R. Chraplyvy, “Optical power limits in multichannel wavelength-division- multiplexed systems due to stimulated Raman scattering,” Electron. Lett. 20, 58–59 (1984).

[CrossRef]

D. E. Pelinovsky, Y. S. Kivshar, and V. V. Afanasjev, “Internal modes of envelope solitons,” Physica D 116, 121–142 (1998).

[CrossRef]

A. Peleg, M. Chertkov, and I. Gabitov, “Inelastic interchannel collisions of pulses in optical fibers in the presence of third-order dispersion,” J. Opt. Soc. Am. B 21, 18–23 (2004).

[CrossRef]

A. Peleg, M. Chertkov, and I. Gabitov, “Interchannel interaction of optical solitons,” Phys. Rev. E 68, 026605 (2003).

[CrossRef]

M. Chertkov, Y. Chung, A. Dyachenko, I. Gabitov, I. Kolokolov, and V. Lebedev, “Shedding and interaction of solitons in weakly disordered optical fibers,” Phys. Rev. E 67, 036615 (2003).

[CrossRef]

F. Forghieri, R. W. Tkach, and A. R. Chraplyvy, “Effect of modulation statistics on Raman crosstalk in WDM systems,” IEEE Photon. Technol. Lett. 7, 101–103 (1995).

[CrossRef]

A. R. Chraplyvy, “Optical power limits in multichannel wavelength-division- multiplexed systems due to stimulated Raman scattering,” Electron. Lett. 20, 58–59 (1984).

[CrossRef]

Y. Chung and A. Peleg, “Monte Carlo simulations of pulse propagation in massive multichannel optical fiber communication systems,” Phys. Rev. A 77, 063835 (2008).

[CrossRef]

Y. Chung and A. Peleg, “Strongly non-Gaussian statistics of optical soliton parameters due to collisions in the presence of delayed Raman response,” Nonlinearity 18, 1555–1574 (2005).

[CrossRef]

A. Peleg and Y. Chung, “Stationary solutions to the nonlinear Schrödinger equation in the presence of third order dispersion,” J. Phys. A 36, 10039–10051 (2003).

[CrossRef]

M. Chertkov, Y. Chung, A. Dyachenko, I. Gabitov, I. Kolokolov, and V. Lebedev, “Shedding and interaction of solitons in weakly disordered optical fibers,” Phys. Rev. E 67, 036615 (2003).

[CrossRef]

G. Genty, J. M. Dudley, and B. J. Eggleton, “Modulation control and spectral shaping of optical fiber supercontinuum generation in the picosecond regime,” Appl. Phys. B 94, 187–194 (2009).

[CrossRef]

M. Chertkov, Y. Chung, A. Dyachenko, I. Gabitov, I. Kolokolov, and V. Lebedev, “Shedding and interaction of solitons in weakly disordered optical fibers,” Phys. Rev. E 67, 036615 (2003).

[CrossRef]

G. Genty, J. M. Dudley, and B. J. Eggleton, “Modulation control and spectral shaping of optical fiber supercontinuum generation in the picosecond regime,” Appl. Phys. B 94, 187–194 (2009).

[CrossRef]

F. Forghieri, R. W. Tkach, and A. R. Chraplyvy, “Effect of modulation statistics on Raman crosstalk in WDM systems,” IEEE Photon. Technol. Lett. 7, 101–103 (1995).

[CrossRef]

A. Peleg, M. Chertkov, and I. Gabitov, “Inelastic interchannel collisions of pulses in optical fibers in the presence of third-order dispersion,” J. Opt. Soc. Am. B 21, 18–23 (2004).

[CrossRef]

A. Peleg, M. Chertkov, and I. Gabitov, “Interchannel interaction of optical solitons,” Phys. Rev. E 68, 026605 (2003).

[CrossRef]

M. Chertkov, Y. Chung, A. Dyachenko, I. Gabitov, I. Kolokolov, and V. Lebedev, “Shedding and interaction of solitons in weakly disordered optical fibers,” Phys. Rev. E 67, 036615 (2003).

[CrossRef]

G. Genty, J. M. Dudley, and B. J. Eggleton, “Modulation control and spectral shaping of optical fiber supercontinuum generation in the picosecond regime,” Appl. Phys. B 94, 187–194 (2009).

[CrossRef]

Y. Kodama and A. Hasegawa, “Nonlinear pulse propagation in a monomode dielectric guide,” IEEE J. Quantum Electron. 23, 510–524 (1987).

[CrossRef]

A. Hause and F. Mitschke, “Reduced soliton interaction by Raman self-frequency-shift,” Phys. Rev. A 80, 063824 (2009).

[CrossRef]

D. E. Pelinovsky, Y. S. Kivshar, and V. V. Afanasjev, “Internal modes of envelope solitons,” Physica D 116, 121–142 (1998).

[CrossRef]

Y. Kodama and A. Hasegawa, “Nonlinear pulse propagation in a monomode dielectric guide,” IEEE J. Quantum Electron. 23, 510–524 (1987).

[CrossRef]

M. Chertkov, Y. Chung, A. Dyachenko, I. Gabitov, I. Kolokolov, and V. Lebedev, “Shedding and interaction of solitons in weakly disordered optical fibers,” Phys. Rev. E 67, 036615 (2003).

[CrossRef]

E. A. Kuznetsov, A. V. Mikhailov, and I. A. Shimokhin, “Nonlinear interaction of solitons and radiation,” Physica D 87, 201–215 (1995).

[CrossRef]

M. Chertkov, Y. Chung, A. Dyachenko, I. Gabitov, I. Kolokolov, and V. Lebedev, “Shedding and interaction of solitons in weakly disordered optical fibers,” Phys. Rev. E 67, 036615 (2003).

[CrossRef]

B. A. Malomed, “Radiative losses in soliton-soliton collisions in an optical fiber with the third-order dispersion,” Phys. Rev. A 43, 3114–3116 (1991).

[CrossRef]
[PubMed]

B. A. Malomed, “Soliton-collision problem in the nonlinear Schrödinger equation with a nonlinear damping term,” Phys. Rev. A 44, 1412–1414 (1991).

[CrossRef]
[PubMed]

E. A. Kuznetsov, A. V. Mikhailov, and I. A. Shimokhin, “Nonlinear interaction of solitons and radiation,” Physica D 87, 201–215 (1995).

[CrossRef]

A. Hause and F. Mitschke, “Reduced soliton interaction by Raman self-frequency-shift,” Phys. Rev. A 80, 063824 (2009).

[CrossRef]

M. Muktoyuk and S. Kumar, “Noise variance due to stimulated Raman scattering among channels of a wavelength-division-multiplexed system,” IEEE Photon. Technol. Lett. 15, 1222–1224 (2003).

[CrossRef]

A. Peleg, “Energy exchange in fast optical soliton collisions as a random cascade model,” Phys. Lett. A 373, 2734–2738 (2009).

[CrossRef]

Y. Chung and A. Peleg, “Monte Carlo simulations of pulse propagation in massive multichannel optical fiber communication systems,” Phys. Rev. A 77, 063835 (2008).

[CrossRef]

A. Peleg, “Intermittent dynamics, strong correlations, and bit-error-rate in multichannel optical fiber communication systems,” Phys. Lett. A 360, 533–538 (2007).

[CrossRef]

Y. Chung and A. Peleg, “Strongly non-Gaussian statistics of optical soliton parameters due to collisions in the presence of delayed Raman response,” Nonlinearity 18, 1555–1574 (2005).

[CrossRef]

J. Soneson and A. Peleg, “Effect of quintic nonlinearity on soliton collisions in optical fibers,” Physica D 195, 123–140 (2004).

[CrossRef]

A. Peleg, M. Chertkov, and I. Gabitov, “Inelastic interchannel collisions of pulses in optical fibers in the presence of third-order dispersion,” J. Opt. Soc. Am. B 21, 18–23 (2004).

[CrossRef]

A. Peleg, “Log-normal distribution of pulse amplitudes due to Raman cross talk in wavelength division multiplexing soliton transmission,” Opt. Lett. 29, 1980–1982 (2004).

[CrossRef]
[PubMed]

A. Peleg, M. Chertkov, and I. Gabitov, “Interchannel interaction of optical solitons,” Phys. Rev. E 68, 026605 (2003).

[CrossRef]

A. Peleg and Y. Chung, “Stationary solutions to the nonlinear Schrödinger equation in the presence of third order dispersion,” J. Phys. A 36, 10039–10051 (2003).

[CrossRef]

D. E. Pelinovsky, Y. S. Kivshar, and V. V. Afanasjev, “Internal modes of envelope solitons,” Physica D 116, 121–142 (1998).

[CrossRef]

E. A. Kuznetsov, A. V. Mikhailov, and I. A. Shimokhin, “Nonlinear interaction of solitons and radiation,” Physica D 87, 201–215 (1995).

[CrossRef]

J. Soneson and A. Peleg, “Effect of quintic nonlinearity on soliton collisions in optical fibers,” Physica D 195, 123–140 (2004).

[CrossRef]

F. Forghieri, R. W. Tkach, and A. R. Chraplyvy, “Effect of modulation statistics on Raman crosstalk in WDM systems,” IEEE Photon. Technol. Lett. 7, 101–103 (1995).

[CrossRef]

H. Yoshida, “Construction of higher order symplectic integrators,” Phys. Lett. A 150, 262–268 (1990).

[CrossRef]

G. Genty, J. M. Dudley, and B. J. Eggleton, “Modulation control and spectral shaping of optical fiber supercontinuum generation in the picosecond regime,” Appl. Phys. B 94, 187–194 (2009).

[CrossRef]

A. R. Chraplyvy, “Optical power limits in multichannel wavelength-division- multiplexed systems due to stimulated Raman scattering,” Electron. Lett. 20, 58–59 (1984).

[CrossRef]

Y. Kodama and A. Hasegawa, “Nonlinear pulse propagation in a monomode dielectric guide,” IEEE J. Quantum Electron. 23, 510–524 (1987).

[CrossRef]

F. Forghieri, R. W. Tkach, and A. R. Chraplyvy, “Effect of modulation statistics on Raman crosstalk in WDM systems,” IEEE Photon. Technol. Lett. 7, 101–103 (1995).

[CrossRef]

M. Muktoyuk and S. Kumar, “Noise variance due to stimulated Raman scattering among channels of a wavelength-division-multiplexed system,” IEEE Photon. Technol. Lett. 15, 1222–1224 (2003).

[CrossRef]

A. Peleg and Y. Chung, “Stationary solutions to the nonlinear Schrödinger equation in the presence of third order dispersion,” J. Phys. A 36, 10039–10051 (2003).

[CrossRef]

Y. Chung and A. Peleg, “Strongly non-Gaussian statistics of optical soliton parameters due to collisions in the presence of delayed Raman response,” Nonlinearity 18, 1555–1574 (2005).

[CrossRef]

M. H. Frosz, O. Bang, and A. Bjarklev, “Soliton collision and Raman gain regimes in continuous-wave pumped supercontinuum generation,” Opt. Express 14, 9391–9407 (2006).

[CrossRef]
[PubMed]

N. Korneev, E. A. Kuzin, B. Ibarra-Escamilla, M. Bello-Jiménez, and A. Flores-Rosas, “Initial development of supercontinuum in fibers with anomalous dispersion pumped by nanosecond-long pulses,” Opt. Express 16, 2636–2645 (2008).

[CrossRef]
[PubMed]

M. Olivier, V. Roy, and M. Pichè, “Influence of the Raman effect on bound states of dissipative solitons,” Opt. Express 14, 9728–9742 (2006).

[CrossRef]
[PubMed]

F. Luan, D. V. Skryabin, A. V. Yulin, and J. C. Knight, “Energy exchange between colliding solitons in photonic crystal fibers,” Opt. Express 14, 9844–9853 (2006).

[CrossRef]
[PubMed]

S. Chi and S. Wen, “Raman cross talk of soliton collision in a lossless fiber,” Opt. Lett. 14, 1216–1218 (1989).

[CrossRef]
[PubMed]

S. Kumar, “Influence of Raman effects in wavelength-division multiplexed soliton systems,” Opt. Lett. 23, 1450–1452 (1998).

[CrossRef]

A. Peleg, “Log-normal distribution of pulse amplitudes due to Raman cross talk in wavelength division multiplexing soliton transmission,” Opt. Lett. 29, 1980–1982 (2004).

[CrossRef]
[PubMed]

A. Efimov, A. J. Taylor, F. G. Omenetto, and E. Vanin, “Adaptive control of femtosecond soliton self-frequency shift in fibers,” Opt. Lett. 29, 271–273 (2004).

[CrossRef]
[PubMed]

F. M. Mitschke and L. F. Mollenauer, “Discovery of the soliton self-frequency shift,” Opt. Lett. 11, 659–661 (1986).

[CrossRef]
[PubMed]

J. P. Gordon, “Theory of the soliton self-frequency shift,” Opt. Lett. 11, 662–664 (1986).

[CrossRef]
[PubMed]

T. I. Lakoba and D. J. Kaup, “Influence of the Raman effect on dispersion-managed solitons and their interchannel collisions,” Opt. Lett. 24, 808–810 (1999).

[CrossRef]

A. Peleg, “Energy exchange in fast optical soliton collisions as a random cascade model,” Phys. Lett. A 373, 2734–2738 (2009).

[CrossRef]

A. Peleg, “Intermittent dynamics, strong correlations, and bit-error-rate in multichannel optical fiber communication systems,” Phys. Lett. A 360, 533–538 (2007).

[CrossRef]

H. Yoshida, “Construction of higher order symplectic integrators,” Phys. Lett. A 150, 262–268 (1990).

[CrossRef]

D. J. Kaup, “Second-order perturbations for solitons in optical fibers,” Phys. Rev. A 44, 4582–4590 (1991).

[CrossRef]
[PubMed]

B. A. Malomed, “Radiative losses in soliton-soliton collisions in an optical fiber with the third-order dispersion,” Phys. Rev. A 43, 3114–3116 (1991).

[CrossRef]
[PubMed]

B. A. Malomed, “Soliton-collision problem in the nonlinear Schrödinger equation with a nonlinear damping term,” Phys. Rev. A 44, 1412–1414 (1991).

[CrossRef]
[PubMed]

Y. Chung and A. Peleg, “Monte Carlo simulations of pulse propagation in massive multichannel optical fiber communication systems,” Phys. Rev. A 77, 063835 (2008).

[CrossRef]

A. Hause and F. Mitschke, “Reduced soliton interaction by Raman self-frequency-shift,” Phys. Rev. A 80, 063824 (2009).

[CrossRef]

M. Chertkov, Y. Chung, A. Dyachenko, I. Gabitov, I. Kolokolov, and V. Lebedev, “Shedding and interaction of solitons in weakly disordered optical fibers,” Phys. Rev. E 67, 036615 (2003).

[CrossRef]

A. Peleg, M. Chertkov, and I. Gabitov, “Interchannel interaction of optical solitons,” Phys. Rev. E 68, 026605 (2003).

[CrossRef]

E. A. Kuznetsov, A. V. Mikhailov, and I. A. Shimokhin, “Nonlinear interaction of solitons and radiation,” Physica D 87, 201–215 (1995).

[CrossRef]

J. Soneson and A. Peleg, “Effect of quintic nonlinearity on soliton collisions in optical fibers,” Physica D 195, 123–140 (2004).

[CrossRef]

D. E. Pelinovsky, Y. S. Kivshar, and V. V. Afanasjev, “Internal modes of envelope solitons,” Physica D 116, 121–142 (1998).

[CrossRef]

M.N.Islam, ed., Raman Amplifiers for Telecommunications 1: Physical Principles (Springer, 2004).

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2001).

C.Headley and G.P.Agrawal, eds., Raman Amplification in Fiber Optical Communication Systems (Elsevier, 2005).

The dimensionless z in Eq. is z=(|β2|X)/(2τ02), where X is the actual position, τ0 is the soliton width, and β2 is the second order dispersion coefficient. The dimensionless retarded time is t=τ/τ0, where τ is the retarded time. The spectral width is ν0=1/(π2τ0) and the frequency difference is Δν=(πΔβν0)/2. ψ=E/P0, where E is proportional to the electric field and P0 is the peak power. The dimensionless second order dispersion coefficient is d=−1=β2/(γP0τ02), where γ is the Kerr nonlinearity coefficient. The coefficient ϵR is given by ϵR=0.006/τ0, where τ0 is in picoseconds.