Density-phase conjugacy
= Density-phase conjugacy
In a first-order condensate action, density and phase are canonically paired variables. Linear density and phase oscillations are therefore coupled components of a single Bogoliubov branch, rather than two independent positive-frequency particles. For $i\dot\psi=-\psi''+\beta(|\psi|^2-1)\psi$, writing $\psi\simeq1+\rho/2+i\theta$ gives $\dot\rho=-2\theta''$ and $\dot\theta=\rho''/2-\beta\rho$, hence $\omega^2=k^4+2\beta k^2$.