424 CHAPTER 9. SOLITON SYSTEMS
0 1020304050
Distance (km)0.00.20.40.60.81.01.2Normalized EnergyFigure 9.12: Variations in soliton energy for backward (solid line) and bidirectional (dashed
line) pumping schemes withLA=50 km. The lumped-amplifier case is shown by the dotted
line.
p(z)varies increases withLA. Nevertheless, it remains much smaller than that occur-
ring in the lumped-amplification case. As an example, soliton energy varies by a factor
of 100 or more whenLA=100 km if lumped amplification is used but by less than a
factor of 2 when the bidirectional pumping scheme is used for distributed amplification.
The effect of energy excursion on solitons depends on the ratioξA=LA/LD. When
ξA<1, little soliton reshaping occurs. ForξA1, solitons evolve adiabatically with
some emission of dispersive waves (the quasi-adiabatic regime). For intermediate val-
ues ofξA, a more complicated behavior occurs. In particular, dispersive waves and
solitons are resonantly amplified whenξA 4 π. Such a resonance can lead to unsta-
ble and chaotic behavior [60]. For this reason, distributed amplification is used with
ξA< 4 πin practice [62]–[66].
Modeling of soliton communication systems making use of distributed amplifica-
tion requires the addition of a gain term to the NLS equation, as in Eq. (9.3.4). In the
case of soliton systems operating at bit ratesB>20 Gb/s such thatT 0 <5 ps, it is also
necessary to include the effects of third-order dispersion (TOD) and a new nonlinear
phenomenon known as thesoliton self-frequency shift(SSFS). This effect was discov-
ered in 1986 [67] and can be understood in terms of intrapulse Raman scattering [68].
The Raman effect leads to a continuous downshift of the soliton carrier frequency when
the pulse spectrum becomes so broad that the high-frequency components of a pulse
can transfer energy to the low-frequency components of the same pulse through Ra-
man amplification. The Raman-induced frequency shift is negligible forT 0 >10 ps but
becomes of considerable importance for short solitons (T 0 <5 ps). With the inclusion
of SSFS and TOD, Eq. (9.3.4) takes the form [10]
i∂u
∂ξ+
1
2
∂^2 u
∂τ^2+|u|^2 u=iLD
2[g(ξ)−α]u+iδ 3∂^3 u
∂τ^3+τRu∂|u|^2
∂τ