= Localized pulse of a conserved-field convection model
{title2=$A=R\operatorname{sech}(cX)$}
For a real steady amplitude and zero-mean field, twice integrating the field equation yields $B=\kappa(\langle A^2\rangle-A^2)$, $\kappa=\mu/\sigma$. The <amplitude equation> becomes $A''+(1-\kappa\langle A^2\rangle)A+(\kappa-1)A^3=0$. Using $(R\operatorname{sech}cX)''=c^2A-2c^2A^3/R^2$ gives $\kappa\langle A^2\rangle=1+c^2$ and $R^2=2c^2/(\kappa-1)$, requiring $\kappa>1$. Its finite-window mean is $2R^2\tanh(cL/2)/(cL)$. For $cL\gg1$, consistency gives $c+c^{-1}=4\kappa/[L(\kappa-1)]$. The leading roots coalesce at $c=1$; for $L>2$ the leading upper parameter limit is $\kappa\leq L/(L-2)$, while retaining the finite-window factor makes the necessary inequality strict. The isolated sech profile has an exponentially small derivative mismatch on a periodic interval, so exact periodic tails need correction. For $L\leq2$ the algebra supplies no finite upper bound of this form.
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