 Hexagonal convection amplitude equations

A finite-wavenumber convection instability with a resonant three-wave triad can yield three coupled amplitude equations. For the gradient nonlinearity $\nabla \cdot (|\nabla \Theta {|}^2\nabla \Theta )$ and weak quadratic asymmetry $-ϵ\gamma \nabla \cdot (\Theta \nabla \Theta )$, projection onto a triad of length squared $K$ gives the stated equation and its cyclic copies. The quadratic coefficient is $+\gamma K$, while both self and cross cubic coefficients are $-3K^2$ for angles of $120$ degrees. The finite critical mode requires positive linear damping in the underlying long-wave equation.