9.1. FIBER SOLITONS 409
Figure 9.3: Pulse evolution for a “sech” pulse withN= 1 .2 over the rangeξ=0–10. The pulse
evolves toward the fundamental soliton (N=1) by adjusting its width and peak power.
9.1.3 Dark Solitons
The NLS equation can be solved with the inverse scattering method even in the case
of normal dispersion [16]. The intensity profile of the resulting solutions exhibits a dip
in a uniform background, and it is the dip that remains unchanged during propagation
inside the fiber [17]. For this reason, such solutions of the NLS equation are called
dark solitons. Even though dark solitons were discovered in the 1970s, it was only
after 1985 that they were studied thoroughly [18]–[28].
The NLS equation describing dark solitons is obtained from Eq. (9.1.5) by changing
the sign of the second term. The resulting equation can again be solved by postulating
a solution in the form of Eq. (9.1.8) and following the procedure outlined there. The
general solution can be written as [28]
ud(ξ,τ)=(ηtanhζ−iκ)exp(iu^20 ξ), (9.1.12)where
ζ=η(τ−κξ), η=u 0 cosφ, κ=u 0 sinφ. (9.1.13)
Here,u 0 is the amplitude of the continuous-wave (CW) background andφis an internal
phase angle in the range 0 toπ/2.
An important difference between the bright and dark solitons is that the speed of a
dark soliton depends on its amplitudeηthroughφ.Forφ=0, Eq. (9.1.12) reduces to
ud(ξ,τ)=u 0 tanh(u 0 τ)exp(iu^20 ξ). (9.1.14)The peak power of the soliton drops to zero at the center of the dip only in theφ= 0
case. Such a soliton is called theblacksoliton. Whenφ=0, the intensity does not
drop to zero at the dip center; such solitons are referred to as thegraysoliton. Another