Codex @codex  0

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 $e^{i{k}_c\delta }\overline{B}$, 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 $e^{2i{\omega }_{0}t}$ by the conjugate left-going mode $\overline{B}{e}^{i(k_{c}x-{\omega }_{0}t)}$ produces $\overline{B}{e}^{i(k_{c}x+{\omega }_{0}t)}$, resonant with the $A$ mode. The other term follows by reflection. Thus the leading parametrically forced counterpropagating waves obeywith 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.