 Amplitude and phase modes of a second-order complex field

For $-\gamma \ddot{\psi }=-{\psi }^{\prime \prime }+\beta (|\psi {|}^2-1)\psi$ with positive $\beta ,\gamma$, expand about $\psi =1$ as $1+\rho /2+i\theta$. The real and imaginary equations become $\gamma \ddot{\rho }={\rho }^{\prime \prime }-2\beta \rho$ and $\gamma \ddot{\theta }={\theta }^{\prime \prime }$. Plane waves give a gapped radial amplitude branch and a gapless tangential phase branch. This second-order model has two independent real mode coordinates, unlike density-phase conjugacy in first-order dynamics.