430 CHAPTER 9. SOLITON SYSTEMS
Numerical solutions, although essential, do not lead to much physical insight. Sev-
eral techniques have been used to solve the NLS equation (9.4.5) approximately. A
common approach makes use of the variational method [89]–[91]. Another approach
expandsB(z,t)in terms of a complete set of the Hermite–Gauss functions that are solu-
tions of the linear problem [92]. A third approach solves an integral equation, derived
in the spectral domain using perturbation theory [94]–[96].
To simplify the following discussion, we focus on the variational method used ear-
lier in Section 7.8.2. In fact, the Lagrangian density obtained there can be used directly
for DM solitons as well as Eq. (9.4.5) is identical to Eq. (7.8.4). Because the shape
of the DM soliton is close to a Gaussian pulse in numerical simulations, the soliton is
assumed to evolve as
B(z,t)=aexp[−( 1 +iC)t^2 / 2 T^2 +iφ], (9.4.6)whereais the amplitude,Tis the width,Cis the chirp, andφis the phase of the
soliton. All four parameters vary withzbecause of perturbations produced by periodic
variations ofβ 2 (z)andp(z).
Following Section 7.8.2, we can obtain four ordinary differential equations for the
four soliton parameters. The amplitude equation can be eliminated becausea^2 T=
a^20 T 0 =E 0 /
√
πis independent ofzand is related to the input pulse energyE 0. The
phase equation can also be dropped sinceTandCdo not depend onφ.TheDM
soliton then corresponds to a periodic solution of the following two equations for the
pulse widthTand chirpC:
dT
dz=β 2 (z)C
T
, (9.4.7)
dC
dz=
γE 0 p(z)
√
2 πT+
β 2
T^2( 1 +C^2 ). (9.4.8)
These equations should be solved with the periodic boundary conditions
T 0 ≡T( 0 )=T(LA), C 0 ≡C( 0 )=C(LA) (9.4.9)to ensure that the soliton recovers its initial state after each amplifier. The periodic
boundary conditions fix the values of the initial widthT 0 and the chirpC 0 atz= 0
for which a soliton can propagate in a periodic fashion for a given value of the pulse
energyE 0. A new feature of the DM solitons is that the input pulse width depends
on the dispersion map and cannot be chosen arbitrarily. In fact,T 0 cannot be below a
critical value that is set by the map itself.
Figure 9.15 shows how the pulse widthT 0 and the chirpC 0 of allowed periodic so-
lutions vary with input pulse energy for a specific dispersion map. The minimum value
Tmof the pulse width occurring in the middle of the anomalous-GVD section of the
map is also shown. The map is suitable for 40-Gb/s systems and consists of alternating
fibers with GVD of−4 and 4 ps^2 /km and lengthsla≈ln=5 km such that the average
GVD is− 0 .01 ps^2 /km. The solid lines show the case of ideal distributed amplifica-
tion for whichp(z)=1 in Eq. (9.4.8). The lumped-amplification case is shown by the
dashed lines in Fig. 9.15 assuming 80-km amplifier spacing and 0.25 dB/km losses in
each fiber section.