= Counterpropagating Hopf amplitudes in rotating convection
{title2=$\dot Z_\pm=rZ_\pm-(g_s|Z_\pm|^2+g_c|Z_\mp|^2)Z_\pm$}
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 $a_s=\operatorname{Re}g_s$ and $a_c=\operatorname{Re}g_c$. 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.
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