Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

1/F
η
0
−1/F
0
a b
m−1 η
gr
gb
c d
π 2π
−p = F t
6.4. Numerical examples
0
k − p
Figure 6.5.: Left panel: Time evolution scheme of stable Bloch oscillations. The inverse
mass m −1 , the interaction parameter of the rigid soliton (6.11) as well as that of a
breathing soliton gb(t) = g0 cos(F t) are shown together with the bounded time η =
sin(F t)/F as function of time or momentum −p = F t. Right panel: The k-space
density [obtained by numerical integration of Eq. (6.5)] of a breathing wave packet is a
function of η and thus strictly periodic in t: the points in time a and b as well as c and
d show the same distribution, respectively.
Again, the time reversal argument applies. Equation (6.15) is solved as
function of the bounded time η ′ , which implies that the wave packet is
conserved under modulations given as
g(t) = sin(τ)P (cos(τ)) = �
gn sin(nτ) = �
n
n
�
n � π
gn sin F t − (2j + 1)�� .
ν2 2
(6.16)
With a different but equivalent derivation, we have published this result in
[129].
6.4. Numerical examples
So far, we have made the following predictions on the basis of the continuous
nonlinear Schrödinger equation (6.8) with time-dependent mass and
interaction:
• The rigid soliton: With an initial state of soliton shape (6.10) and
an interaction parameter modulated as g(t) = −|gr| cos(F t), the wave
packet should completely conserve its shape.
• Modulations g(t) that fulfill the simple time-reversal condition (6.12)
or the generalized one (6.16) are predicted to lead to periodic solutions,
123