 Counterpropagating Hopf amplitudes in rotating convection

Near an oscillatory onset of rotating Rayleigh-Bénard convection, opposite travelling roll waves have coupled Hopf bifurcation amplitudes. Symmetry gives the displayed cubic amplitude equations, with generally complex self-coupling and cross-coupling coefficients. Write $a_s=\mathrm{Re}{g}_s$ and $a_c=\mathrm{Re}{g}_c$. For $r>0,a_s>0$, a single travelling wave has intensity $r/a_s$ and is stable against its counterpropagating competitor when $a_c>a_s$. Equal standing-wave intensities are $r/(a_s+a_c)$; with $r>0$ and positive denominator they are stable to intensity differences when $a_s>a_c$. Phase symmetries give orbital, rather than strict phase-decay, stability. Spatial modulation, mean flows and oblique convection rolls require additional stability tests.