= Weighted energy estimate for two coupled modes
{title2=$N=c|A|^2+a|B|^2$}
Suppose two complex amplitudes satisfy $\partial_t|A|^2\leq2a|A||B|-2d|A|^2$ and $\partial_t|B|^2\leq2c|A||B|-2d|B|^2$, with positive $a,c$ and $d\geq0$. Then $N=c|A|^2+a|B|^2\geq2\sqrt{ac}|A||B|$ gives $\dot N\leq2(\sqrt{ac}-d)N$. A positive weighted norm avoids treating two coupling rates of different size as a single arithmetic sum. This is useful for nonautonomous <linear ordinary differential equations> and the <bounded-modulation alpha-Omega growth estimate>.
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