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 $i\dot{\psi }=-{\psi }^{\prime \prime }+\beta (|\psi {|}^2-1)\psi$, writing $\psi \simeq 1+\rho /2+i\theta$ gives $\dot{\rho }=-2{\theta }^{\prime \prime }$ and $\dot{\theta }={\rho }^{\prime \prime }/2-\beta \rho$, hence ${\omega }^2=k^4+2\beta k^2$.