Convection amplitude coupled to a conserved field

A representative scaled amplitude model has complex $A$ and real $B$ withPeriodic boundary conditions conserve $\int BdX$, because the right-hand side is a total second derivative. The slow field can change long-wave convection roll stability and permit localized pulses. For zero mean $B$, the uniform convection rolls are $A=R{e}^{iqX}$, $R^2=1-q^2$, $B=0$.

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.

Linearize around $A=R{e}^{iqX}$ and write $r=R^2>0$. For sideband wavenumber $l$, the sum and difference of the two amplitude sidebands, together with the field amplitude $b$, evolve underExpanding $\det (\lambda I-M_l)$ givesAt $l=0$, the eigenvalues are $-2r,0,0$; the field's constant mode must be removed if its mean is prescribed. Nonzero long-wave field modes are still dynamically active.

Putting $\lambda =l^2\nu$ into the sideband cubic and retaining $l^4$ givesFor $r=1-q^2>0$, $\mu /\sigma >(1-3q^2)/(1-q^2)$ makes the constant coefficient negative. There is a positive real $\nu$, so sufficiently small nonzero $l$ grows. On a finite period $L$, however, only $l=2\pi j/L$ are allowed. For $q=0$, $\mu =2$, $\sigma =1$, $L=1$, every nonzero allowed mode has negative growth rates despite this long-wave inequality. Thus an arbitrarily-small-wave-number conclusion cannot be asserted without the domain-size qualification.