= Explosive two-to-one amplitude system
{title2=$A_T=-\overline AB/2,\ B_T=-A^2/4$}
For $A_T=-\overline AB/2$ and $B_T=-A^2/4$, write $A=Re^{i\theta}$, $B=Pe^{i\phi}$ and $\delta=\phi-2\theta$. The polar equations are
$$
R_T=-\frac{RP}2\cos\delta,\qquad \theta_T=-\frac P2\sin\delta,\qquad P_T=-\frac{R^2}4\cos\delta,\qquad \phi_T=\frac{R^2}{4P}\sin\delta.
$$
Where the polar phases are defined, direct <differentiation> gives the <first integrals> $R^2-2P^2$ and $R^2P\sin\delta$. In particular $R^2\theta_T$ and $P^2\phi_T$ are constant. If $\delta=\pi$ and $R^2-2P^2=C>0$, then $P_T=P^2/2+C/4$, a <Riccati equation> whose positive solution develops a finite-time pole. This is <finite-time blowup> of the reduced amplitude system; it does not establish blowup of the full wave equation beyond the domain of the <weakly nonlinear expansion>.
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