DW is unaffected; [[i.e., ]{}]{}$\varphi_{\boldsymbol{k}}=\varphi_{0}\exp(-{\rm i}\Omega_{\boldsymbol{k}}t$), with $\varphi_{0}$ being a constant. Then, from (\[eq:Linear\_Primary\_Fourier\]), the equations that describe the ZF and the sidebands are as follows [@St-Onge17]: $$\begin{aligned}
\label{eq:Linear_Secondary_4MT}
{\rm d}_{t}\varphi_{\boldsymbol{q}}=\frac{k_{y}{\rm e}^{\gamma_{\boldsymbol{k}} t}}{q_{x}}\left[\left(q_{x}^{2}-{\rm i}\delta_{+}\right)\varphi_{\boldsymbol{k}_{+}}\varphi_{0}^{*}{\rm e}^{{\rm i}\omega_{\boldsymbol{k}}t}-\left(q_{x}^{2}+{\rm i}\delta_{+}\right)\varphi_{\boldsymbol{k}_{-}}^{*}\varphi_{0}{\rm e}^{-{\rm i}\omega_{\boldsymbol{k}}t}\right],\label{eq:Linear_Secondary_4MT-a}\\
{\rm d}_{t}\varphi_{\boldsymbol{k}_{+}}=-{\rm i}\Omega_{\boldsymbol{k}_{+}}\varphi_{\boldsymbol{k}_{+}}+T(\boldsymbol{k}_{+},\boldsymbol{k},\boldsymbol{q})\varphi_{0}\varphi_{\boldsymbol{q}}{\rm e}^{-{\rm i}\Omega_{\boldsymbol{k}}t},\\
{\rm d}_{t}\varphi_{\boldsymbol{k}_{-}}=-{\rm i}\Omega_{\boldsymbol{k}_{-}}\varphi_{\boldsymbol{k}_{-}}+T(\boldsymbol{k}_{-},\boldsymbol{k},-\boldsymbol{q})\varphi_{0}\varphi_{\boldsymbol{q}}^{*}{\rm e}^{-{\rm i}\Omega_{\boldsymbol{k}}t},\end{aligned}$$ where $\delta_{+}\doteq\delta_{\boldsymbol{k}}+\delta_{\boldsymbol{k}_{+}}=2\delta_{0}k_{y}$. We have also used $\Omega_{\boldsymbol{k}}=\omega_{\boldsymbol{k}}+{\rm i}\gamma_{\boldsymbol{k}}$. These equations can be combined to yield a single time-evolution equation for the ZF amplitude $\varphi_{\boldsymbol{q}}$: $$\frac{{\rm d}^{3}\varphi_{\boldsymbol{q}}}{{\rm d}t^{3}}-A\,\frac{{\rm d}^{2}\varphi_{\boldsymbol{q}}}{{\rm d}t^{2}}+(B-C)\,\frac{{\rm d}\varphi_{\boldsymbol{q}}}{{\rm d}t}-D\varphi_{\boldsymbol{q}}=0.\label{eq:Linear_Secondary_phiqt}$$ Here, $A=2\gamma_{+}$, $B=\omega_{-}^{2}+\gamma_{+}^{2}$, $C,D\propto|\varphi{}_{0}{\rm e}^{\gamma_{\boldsymbol{k}}t}|^{2}$, $\gamma_{+}\doteq\gamma_{\boldsymbol{k}}+\gamma_{\boldsymbol{k}_{+}}$, and $\omega_{-}\doteq\omega_{\boldsymbol{k}}-\omega_{\boldsymbol{k}_{+}}$. The derivation of (\[eq:Linear\_Secondary\_phiqt\]) can be found in @St-Onge17. Expressions for $C$ and $D$ can also be found there but will not be important for our discussion; but note that compared to @St-Onge17, we have absorbed the coefficient ${\rm e}^{\gamma_{\boldsymbol{k}}t}$ into the definitions of $C$ and $D$.

When $C$ and $D$ are much larger than $A$ and $B$, $\varphi_{\boldsymbol{q}}$ can grow “super-exponentially” [@Rogers00; @St-Onge17], [[i.e., ]{}]{}as an exponential of an exponential. This is also known as the secondary KH instability [@Rogers00]. In the opposite case, when $A$ and $B$ dominate over $C$ and $D$, the non-constant solution of (\[eq:Linear\_Secondary\_phiqt\]) is approximately $$\varphi_{\boldsymbol{q}}\propto{\rm e}^{(\gamma_{+}\pm{\rm i}\omega_{-})t}.$$ Since $\gamma_{+}$ decreases as $|q_{x}|$ increases (see (\[eq:Linear\_Primary\_Omegak\])), 