Nonlinear Fiber Optics - 4 ed. Agrawal

5.5. Higher-Order Effects 161
1
N = 1
s = 0.2
0.8
Intensity
0.6
0.4
z/L D
= 0
5
0.2
10
0
−5 0 5
Time, T/T 0
Figure 5.18: Pulse shapes at z/L D = 5 and 10 for a fundamental soliton in the presence of selfsteepening
(s = 0.2). The dotted curve shows the initial pulse shape at z = 0 for comparison. The
solid and dashed curves coincide with the dotted curve when s = 0.
its soliton nature. This feature suggests that Eq. (5.5.15) has a soliton solution toward
which the input pulse is evolving asymptotically. Such a solution indeed exists and has
the form [146]
u(ξ ,τ)=V (τ + Mξ )exp[i(Kξ − Mτ)], (5.5.16)
where M is related to a shift Ω p of the carrier frequency. The group velocity changes
as a result of the shift. The delay of the peak seen in Figure 5.18 is due to this change
in the group velocity. The explicit form of V (τ) depends on M and s [183]. In the limit
s = 0, it reduces to the hyperbolic secant form given in Eq. (5.2.16). Note also that Eq.
(5.5.15) can be transformed into a so-called derivative NLS equation that is integrable
by the inverse scattering method and whose solutions have been studied extensively in
plasma physics [184]–[187].
The effect of self-steepening on higher-order solitons is remarkable in that it leads
to breakup of such solitons into their constituents, a phenomenon referred to as soliton
fission [180]. Figure 5.19 shows this behavior for a second-order soliton (N = 2) by
displaying the temporal and spectral evolutions for s = 0.2. For this relatively large
value of s, the two solitons have separated from each other within a distance of 2L D and
continue to move apart with further propagation inside the fiber. A qualitatively similar
behavior occurs for smaller values of s except that a longer distance is required for the
breakup of solitons. The soliton fission can be understood from the inverse scattering
method, with the self-steepening term acting as a perturbation. In the absence of selfsteepening
(s = 0), the two solitons form a bound state because both of them propagate
at the same speed (the eigenvalues ζ j in Section 5.2.1 have the same real part). The
effect of self-steepening is to break the degeneracy so that the two solitons propagate at