= Solution
Forcing at $2\omega_0$ preserves translation by one forcing period, which acts on the critical amplitudes as $(A,B)\mapsto(-A,-B)$. It destroys the continuous common phase symmetry. Under the remaining <spatial translation>, $\overline B$ transforms as $e^{ik_c\delta}\overline B$, just like $A$, and $\overline A$ transforms just like $B$. Both change sign under the remaining temporal sign reversal. They are therefore allowed linear terms in the respective <amplitude equations>. Spatial reflection requires their coefficients to coincide.
The resonant physical-frequency calculation gives the same result: multiplying the forcing factor $e^{2i\omega_0t}$ by the conjugate left-going mode $\overline B e^{i(k_cx-\omega_0t)}$ produces $\overline B e^{i(k_cx+\omega_0t)}$, resonant with the $A$ mode. The other term follows by reflection. Thus the leading <parametrically forced counterpropagating waves> obey
$$
\boxed{A_T=\mu A+\nu\overline B-\beta A|A|^2-\gamma A|B|^2,}
$$
with the reflected equation for $B$. The forcing phase can be absorbed into a common amplitude phase so that $\nu$ is real; its magnitude measures the resonant forcing strength. Higher-order forcing corrections are beyond this leading weak-forcing truncation.
Back to article page
