= Solution
Use the nonzero homogeneous <number density> $n_0=\mu/g$ from the previous part, with $g,\mu>0$. Set
$$
\boxed{\widetilde\psi=\frac\psi{\sqrt{n_0}},\qquad
\widetilde{\mathbf x}=\frac{\mathbf x}{\ell_h},\quad
\ell_h=\frac\hbar{\sqrt{2m\mu}},\qquad
\widetilde t=\frac{2\mu}{\hbar}t.}
$$
The length $\ell_h$ is the conventional <healing length> in this normalization. In the dimensional <Gross–Pitaevskii equation>, $\hbar^2/(2m\ell_h^2)=\mu$ and $g n_0=\mu$. Substitution and division by $\mu\sqrt{n_0}$ therefore give
$$
2i\partial_{\widetilde t}\widetilde\psi=-\widetilde\nabla^2\widetilde\psi+(|\widetilde\psi|^2-1)\widetilde\psi,
$$
or equivalently the requested $-2i$ form. The factor two in the time scale is necessary: using $\mu t/\hbar$ instead would leave a coefficient $-i$. The normalized <number density> is $|\widetilde\psi|^2=n/n_0$.
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