= Long-wave convection equation with broken Boussinesq symmetry
{title2=$\Theta_t=-\Theta-\mu\Theta_{xx}-\Theta_{xxxx}-s(\Theta_x^2)_x+(\Theta_x^3)_x$}
A reduced long-wave <convection> model can combine linear damping, destabilizing second derivatives, stabilizing fourth derivatives and gradient nonlinearities. In the displayed normalization the quadratic coefficient $s$ breaks <Boussinesq symmetry>, while the cubic gradient term dissipates the mean-square temperature. The linear <dispersion relation> is $\lambda(k)=-1+\mu k^2-k^4$, with critical <wavenumber> $k=1$ at $\mu=2$. Smoothness and spatial averaging hypotheses must be specified before applying the <energy method> on an infinite interval.
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