As discussed in Section #GDS#423>, when smaller values of
δ are used, pulses of better
characteristics are obtained. This can be seen in Fig. 1(d), where
#math225#a = 1.33 and a pure fundamental dark soliton is
generated. ....
102
verbatim145#
103
We first examine the solution of a modified NLSE with a constant gain:
#math226#
iuz - 1/2utt + | u|2u = iΓu,
(2)
where Γ is assumed to be a constant,
appropriate for the Raman amplification
under strong pumping without depletion. The
solution of a similar equation to Eq. (#E2#432>),
but ...
#mathletters433#
Under this transformation, the NLSE has the new form
The solution of Eq. (#E2#452>) when Γ = 0 is well known and
has the form #math228#exp[iσ(z, t)]κtanhκt,
where κ is the form factor and the phase variable satisfies
#math229#∂σ/∂z = κ2.[#ZA##1###]
Therefore, when the right-hand-side of
Eq.(#E7#455>) is zero, an exact solution for v(z', t) can be
obtained from Eq. (#E7#456>).
On the other hand, when #math230#z→∞ and hence
#math231#z'→∞ or #math232#Γ→ 0, the
right-hand side of Eq. (#E7#457>) becomes infinitely small.
Under these conditions, we can treat the right-hand
side of Eq. (#E7#458>) as a perturbation to the NLSE.
...