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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.

Because $⟨B⟩=0$, a periodic primitive $H$ exists. Fix its additive constant by $⟨H⟩=0$. Integrating the conserved-field equation once in space gives $H_T=\sigma (H_X+\delta A^2{)}_X$: the possible integration constant vanishes on taking the spatial mean. Treat the spatial average as the inner product defining functional derivatives. Integration by parts, with no boundary terms under periodic boundary conditions, givesThus the conserved-field real amplitude equation is a gradient flow ascending $V$, or descending $-V$, with positive diagonal mobilities. Along a classical solution,For the mixed term put $P=⟨A^4{⟩}^{1/2}$ and $Q=⟨H_X^2{⟩}^{1/2}$. The Cauchy-Schwarz inequality and a completed square giveConsequently, for $\beta =1-\delta >0$,The negative leading quartic coefficient proves the upper bound. The stronger pointwise square completionis also useful: a superlevel set $V\ge V(0)$ bounds the amplitude in $L^4$, its spatial derivative in $L^2$, and $B=H_X$ in $L^2$, because a quartic polynomial with negative leading coefficient dominates its lower-order terms.

The dynamics dissipates toward the steady-state set rather than sustaining nonconstant periodic motion. Indeed $V$ increases to a finite limit, and the time integral of $⟨A_T^2+H_T^2/(2\delta \sigma )⟩$ is finite. For global regular trajectories with the usual parabolic precompactness, every limit point is stationary, by the LaSalle invariance principle. A nonconstant periodic or recurrent orbit would force a strictly increasing $V$ to return to its original value and is impossible. Boundedness of a monotone functional alone does not prove convergence to one particular equilibrium; isolation of the limiting equilibrium or an additional convergence theorem supplies that stronger claim.

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.