= Solution
To first order, $|1+u+iv|^2=1+2u$ and $(1-|\psi|^2)\psi=-2u$. Equating real and imaginary parts of the <Gross–Pitaevskii equation> therefore gives
$$
\boxed{2v_t=\nabla^2u-2u,\qquad -2u_t=\nabla^2v}.
$$
Differentiate the second equation in time and use the first to eliminate $v$:
$$
\boxed{u_{tt}=\frac12\nabla^2u-\frac14\nabla^4u}.
$$
For a <Fourier mode> with $k=|\mathbf k|$, the <dispersion relation> is
$$
-\omega^2=-\frac{k^2}{2}-\frac{k^4}{4},\qquad
\boxed{\omega_\pm(k)=\pm\frac{k}{2}\sqrt{k^2+2}}.
$$
This is the normalized <Bogoliubov spectrum>. At small <wavenumber> the positive branch has sound speed $\lim_{k\to0}\omega_+(k)/k=1/\sqrt2$; at large <wavenumber> it approaches the free-particle quadratic dispersion. The linear frequencies are real, confirming stability of the repulsive uniform bulk.
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