= Solution
The <dark soliton> density is
$$
|\psi_0(\xi)|^2
=1-(1-2U^2)
\operatorname{sech}^2\left(
\frac{\sqrt{1-2U^2}}{\sqrt2}\xi\right).
$$
For $U=0$,
$$
\boxed{|\psi_0(\xi)|^2=\tanh^2(\xi/\sqrt2)},
$$
a stationary notch reaching zero at its centre. For $U=1/2$,
$$
\boxed{|\psi_0(\xi)|^2
=1-\frac12\operatorname{sech}^2(\xi/2)},
$$
a shallower notch with minimum $1/2$. Since $\xi=x-Ut$, the first profile remains fixed and the second translates rigidly to the right at speed $1/2$.
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