9.1. FIBER SOLITONS 407
Figure 9.1: Evolution of the first-order (left column) and third-order (right column) solitons over
one soliton period. Top and bottom rows show the pulse shape and chirp profile, respectively.
This equation can be solved by multiplying it by 2(dV/dτ)and integrating overτ. The
result is given as
(dV/dτ)^2 = 2 KV^2 −V^4 +C, (9.1.10)
whereCis a constant of integration. Using the boundary condition that bothVand
dV/dτshould vanish at|τ|=∞for pulses,Cis found to be 0. The constantKis de-
termined using the other boundary condition thatV=1 anddV/dτ=0 at the soliton
peak, assumed to occur atτ=0. Its use providesK=^12 , and henceφ=ξ/2. Equa-
tion (9.1.10) is easily integrated to obtainV(τ)=sech(τ). We have thus found the
well-known “sech” solution [11]–[13]
u(ξ,τ)=sech(τ)exp(iξ/ 2 ) (9.1.11)for the fundamental soliton by integrating the NLS equation directly. It shows that the
input pulse acquires a phase shiftξ/2 as it propagates inside the fiber, but its amplitude
remains unchanged. It is this property of a fundamental soliton that makes it an ideal
candidate for optical communications. In essence, the effects of fiber dispersion are
exactly compensated by the fiber nonlinearity when the input pulse has a “sech” shape
and its width and peak power are related by Eq. (9.1.4) in such a way thatN=1.
An important property of optical solitons is that they are remarkably stable against
perturbations. Thus, even though the fundamental soliton requires a specific shape and