= Conserved-field real amplitude equation
{title2=$A_T=\mu A+\alpha A^2-A^3+A_{XX}-AB,\quad B_T=\sigma(B+\delta A^2)_{XX}$}
With <periodic boundary conditions>, the spatial mean of $B$ is a <conserved quantity>. For zero mean write $B=H_X$ with periodic zero-mean $H$. The functional $V=\langle\mu A^2/2+\alpha A^3/3-A^4/4-A_X^2/2-A^2H_X/2-H_X^2/(4\delta)\rangle$ generates $A_T=\delta V/\delta A$, $H_T=2\delta\sigma\,\delta V/\delta H$. Thus $V_T=\langle A_T^2+H_T^2/(2\delta\sigma)\rangle\geq0$. Completing the square in $H_X$ bounds $V$ above by a coercive quartic polynomial when $0<\delta<1$, ruling out nonconstant recurrent motion on regular precompact trajectories.
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