Localized pulse of a conserved-field convection model ($A=R\mathrm{sech}(cX)$)

For a real steady amplitude and zero-mean field, twice integrating the field equation yields $B=\kappa (⟨A^2⟩-A^2)$, $\kappa =\mu /\sigma$. The amplitude equation becomes $A^{\prime \prime }+(1-\kappa ⟨A^2⟩)A+(\kappa -1)A^3=0$. Using $(R\mathrm{sech}cX{)}^{\prime \prime }=c^{2}A-2c^2{A}^3/R^2$ gives $\kappa ⟨A^2⟩=1+c^2$ and $R^2=2c^2/(\kappa -1)$, requiring $\kappa >1$. Its finite-window mean is $2R^2\tanh (cL/2)/(cL)$. For $cL\gg 1$, 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 $\kappa \le L/(L-2)$, 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 $L\le 2$ the algebra supplies no finite upper bound of this form.