= Amplitude and phase modes of a second-order complex field
{title2=$\omega_\rho^2=(k^2+2\beta)/\gamma,\quad\omega_\theta^2=k^2/\gamma$}
For $-\gamma\ddot\psi=-\psi''+\beta(|\psi|^2-1)\psi$ with positive $\beta,\gamma$, expand about $\psi=1$ as $1+\rho/2+i\theta$. The real and imaginary equations become $\gamma\ddot\rho=\rho''-2\beta\rho$ and $\gamma\ddot\theta=\theta''$. 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.
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