Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 2 / 8A / iv / Solution

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For the normalization
$$
-2i\psi_t=\psi_{xx}+(1-|\psi|^2)\psi,
$$
a dark soliton of speed $0\leq U<1/\sqrt2$ is
$$
\psi_0(\xi)=
\sqrt{1-2U^2}\,
\tanh\left(\frac{\sqrt{1-2U^2}}{\sqrt2}\xi\right)
+i\sqrt2U,
\qquad \xi=x-Ut.
$$

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