= Conserved-mean convection amplitude equations
{title2=$A_\tau=\nu A-3|A|^2A-2|B|^2A$}
The <partial differential equation> $T_t=-\mu\Delta T-\Delta^2T+\nabla\cdot(|\nabla T|^2\nabla T)$ on a $2\pi$-periodic square conserves the spatial mean. On a fixed-mean subspace its first instability is at $\mu=1$ and has four real critical <amplitudes>. With $\mu=1+\varepsilon^2\nu$ and $T=\varepsilon(Ae^{ix}+Be^{iy}+\text{complex conjugate})+\cdots$, projection onto the critical <Fourier modes> gives the displayed equation and its $A,B$ interchange. For $\nu>0$, squares have $|A|^2=|B|^2=\nu/5$, <amplitude> <eigenvalues> $-2\nu,-2\nu/5$, and neutral translation <phases>. Rolls have a positive transverse growth rate $\nu/3$.
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