 Convection amplitude coupled to a conserved field

A representative scaled amplitude model has complex $A$ and real $B$ withPeriodic boundary conditions conserve $\int BdX$, because the right-hand side is a total second derivative. The slow field can change long-wave convection roll stability and permit localized pulses. For zero mean $B$, the uniform convection rolls are $A=R{e}^{iqX}$, $R^2=1-q^2$, $B=0$.