Conserved-field real amplitude equation ($A_T=\mu A+\alpha A^2-A^3+A_{XX}-AB,B_T=\sigma (B+\delta A^2{)}_{XX}$)

With periodic boundary conditions, the spatial mean of $B$ is a conserved quantity. For zero mean write $B=H_X$ with periodic zero-mean $H$. The functional $V=⟨\mu A^2/2+\alpha A^3/3-A^4/4-A_X^2/2-A^2{H}_X/2-H_X^2/(4\delta )⟩$ generates $A_T=\delta V/\delta A$, $H_T=2\delta \sigma \delta V/\delta H$. Thus $V_T=⟨A_T^2+H_T^2/(2\delta \sigma )⟩\ge 0$. Completing the square in $H_X$ bounds $V$ above by a coercive quartic polynomial when $0<\delta <1$, ruling out nonconstant recurrent motion on regular precompact trajectories.

For $0<\delta <1$ and $\alpha \ne 0$, a broad localized plateau in the conserved-field real amplitude equation has effective parameter $\nu =\mu -\delta ⟨A^2⟩$ and stationary equation $A^{\prime \prime }+\nu A+\alpha A^2-(1-\delta )A^3=0$. Equal endpoint values of its first integral give $R=2\alpha /[3(1-\delta )]$, $\nu =-2{\alpha }^2/[9(1-\delta )]$. If the plateau occupies fraction $h$ of a long periodic interval, $⟨A^2⟩\simeq h{R}^2$ supplies the displayed relation. It applies when both the plateau and exterior are long relative to the front width; $0<h<1$ gives its leading existence interval.

A nonzero uniform equilibrium of the conserved-field real amplitude equation obeys $\mu =R^2-\alpha R$. Fixed zero mean excludes the constant $B$ perturbation. Its homogeneous amplitude eigenvalue is $-r$. For a nonzero Fourier wavenumber $k$, the stability matrix has trace $-r-(1+\sigma )k^2$ and determinant $\sigma k^2(k^2+r-2\delta R^2)$. The displayed conditions are necessary and sufficient for strict linear stability on a period $2L$. In an infinite-domain limit the nonzero-mode condition becomes $r\ge 2\delta R^2$; a long-wave instability need not occur in a sufficiently short domain.