Travelling and standing convection rolls

At fixed orientation, opposite travelling amplitudes can obey ${\dot{Z}}_{\pm }=(r+i{\omega }_0)Z_{\pm }-(g|Z_{\pm }{|}^2+h|Z_{\mp }{|}^2)Z_{\pm }$. For $r>0$ and $g_r=\mathrm{Re}g>0$, a travelling branch has one nonzero intensity $r/g_r$ and is stable to its counter-propagating amplitude when $h_r>g_r$. A standing branch has both intensities $r/(g_r+h_r)$ and is stable within the pair when $g_r>h_r$ and $g_r+h_r>0$. These follow by linearizing the real intensity equations. Spatial modulation and other orientations remain separate stability tests.