Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 84 / 2 / i / 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 givesDifferentiate the second equation in time and use the first to eliminate $v$:For a Fourier mode with $k=|\mathbf{k}|$, the dispersion relation isThis is the normalized Bogoliubov spectrum. At small wavenumber the positive branch has sound speed ${\lim }_{k\to 0}{\omega }_{+}(k)/k=1/\sqrt{2}$; at large wavenumber it approaches the free-particle quadratic dispersion. The linear frequencies are real, confirming stability of the repulsive uniform bulk.