= Parametrically forced counterpropagating waves
{title2=$A_T=\mu A+\nu\overline B-\beta A|A|^2-\gamma A|B|^2$}
Two counterpropagating modes transform as $(A,B)\mapsto(e^{ik\delta}A,e^{-ik\delta}B)$ under <spatial translation>. Forcing at twice their natural frequency breaks continuous temporal phase symmetry but preserves simultaneous sign reversal. Hence $\overline B$ transforms like $A$ and $\overline A$ like $B$, allowing linear parametric coupling. An equal-amplitude phase-locked <standing wave> has $|A|^2=|B|^2=z>0$ with $|\beta+\gamma|^2z^2-2\mu\operatorname{Re}(\beta+\gamma)z+\mu^2-\nu^2=0$. A nonnegative <discriminant> is only an algebraic compatibility condition: a physical nonzero wave also requires a positive root.
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