 Long-wave convection equation with broken Boussinesq symmetry

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.