Conserved-mean convection amplitude equations ($A_{\tau }=\nu A-3|A{|}^{2}A-2|B{|}^{2}A$)

The partial differential equation $T_t=-\mu \Delta T-{\Delta }^{2}T+\nabla \cdot (|\nabla T{|}^2\nabla T)$ 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 $\mu =1+{\epsilon }^2\nu$ and $T=\epsilon (A{e}^{ix}+B{e}^{iy}+\text{complex conjugate})+\cdots$, 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 $\nu /3$.