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A mesa is a localized broad plateau joined to its surrounding state by two separated fronts. Its leading spatial average equals the plateau fraction times the plateau value; the transition layers give corrections proportional to their width divided by the domain length. A conserved mean can determine the fraction occupied by the plateau, as in the coexistence fraction of a conserved-field amplitude mesa.

At a periodic steady state, two spatial integrations of $0=(B+\delta A^2{)}_{XX}$ make $B+\delta A^2$ constant. The zero-mean constraint fixes that constant to $\delta ⟨A^2⟩$. Define $\beta =1-\delta >0$ and $\nu =\mu -\delta ⟨A^2⟩$; the stationary amplitude equation isFor a broad localized plateau, the two transition layers occupy $O(\ell )$ length while the plateau and exterior occupy $2hL$ and $2(1-h)L$. Provided both lengths are large compared with the front width $\ell$, spatial averaging gives $⟨A^2⟩=h{R}^2+O(R^2\ell /L)$.

Multiplying the stationary equation by $A_X$ gives the first integralIn the well-separated-front approximation, $A\to 0$ with $A_X\to 0$ outside, so $E=0$. On the plateau $A\to R\ne 0$, giving both $\nu +\alpha R-\beta R^2=0$ and $W(R)=0$. Substitution of the first into the second gives $R^3(\beta R/4-\alpha /6)=0$, henceThe equal-potential condition is therefore $2{\alpha }^2=-9\nu \beta$. It is a front balance, not an integration of the stationary equation without its $A_X$ multiplier.

These plateau values also produce actual isolated fronts. At this balance $W(A)=-\beta A^2(A-R{)}^2/4$, so for $\alpha >0$ the increasing front satisfies $A_X=\sqrt{\beta /2}A(R-A)$. It isFor $\alpha <0$ the sign-reversed construction gives the negative plateau. Two oppositely oriented well-separated fronts give the leading periodic mesa solution approximation; their weak interaction and tails supply finite-$L$ corrections.

Finally $\nu =\mu -\delta h{R}^2$ gives the coexistence fraction of a conserved-field amplitude mesa:The leading existence range is the interval obtained from $0<h<1$:This assumes $\alpha \ne 0$ and $L\gg \ell$, with $hL,(1-h)L\gg \ell$. At the two displayed ends a transition pair can no longer be treated as well separated, so they are asymptotic bounds rather than exact finite-period bifurcation values. For $\alpha =0$ this particular nonzero plateau balance collapses to $R=0$ and does not yield the mesa solution family.