is maximized at $(k_{x},k_{y})=(0,1)$. A nonzero $D_{\boldsymbol{k}}$ can modify the value of $\boldsymbol{k}$ that maximizes $\gamma_{\boldsymbol{k}}$, but for the chosen form of $\hat{D}$, (\[eq:Equations\_D\]), this modification is very small. Therefore, if one numerically simulates (\[eq:Equations\_mTHE\]) with small random noise as the initial conditions, then nonlinear interactions can be neglected at first and coherent DW structures will grow exponentially with typical wavenumber $\boldsymbol{k}\approx(0,1)$, as seen in figure \[fig:THhistory\].

Secondary instability {#subsec:Linear_Secondary}
---------------------

When many Fourier modes are present and have grown to a finite amplitude, the nonlinear term in (\[eq:Equations\_mTHE\]) becomes important. This can be seen from the Fourier representation, $\varphi=\sum_{\boldsymbol{k}}\varphi_{\boldsymbol{k}}(t)\exp({\rm i}\boldsymbol{k}\cdot\boldsymbol{x})$, where (\[eq:Equations\_mTHE\]) is written as $$\frac{{\rm d}\varphi_{\boldsymbol{k}}}{{\rm d}t}=-{\rm i}\Omega_{\boldsymbol{k}}\varphi_{\boldsymbol{k}}+\frac{1}{2}\sum_{\boldsymbol{k}_{1},\boldsymbol{k}_{2}}T(\boldsymbol{k},\boldsymbol{k}_{1},\boldsymbol{k}_{2})\delta_{\boldsymbol{k},\boldsymbol{k}_{1}+\boldsymbol{k}_{2}}\varphi_{\boldsymbol{k}_{1}}\varphi_{\boldsymbol{k}_{2}}\label{eq:Linear_Primary_Fourier}$$ and $\delta_{\boldsymbol{k}_1, \boldsymbol{k}_2}$ is the Kronecker symbol. Also, $$T(\boldsymbol{k},\boldsymbol{k}_{1},\boldsymbol{k}_{2})\doteq-\frac{\bar{k}_{1}^{2}-\bar{k}_{2}^{2}}{\bar{k}^{2}}(\boldsymbol{k}_{1}\times\boldsymbol{k}_{2})\cdot\hat{\boldsymbol{z}}$$ are the coefficients that govern the nonlinear mode coupling, $\bar{k}^{2}$ is defined as $$\bar{k}^{2}\doteq\alpha_{\boldsymbol{k}}+k^{2}-{\rm i}\delta_{0}k_{y},\label{eq:Equations_kbar}$$ and similarly for $\bar{k}_{1}^{2}$ and $\bar{k}_{2}^{2}$.

Due to nonlinear interactions, ZFs can be generated from DWs, which process is known as the secondary instability. Here, we use the 4MT model to analyze this instability, namely, by considering a primary DW with $\boldsymbol{k}=(0,k_{y})$, a ZF with $\boldsymbol{q}=(q_{x},0)$, and two DW sidebands with $\boldsymbol{k}_{\pm}=(\pm q_{x},k_{y})$. Assume that the ZF is small, so the exponential growth of the primary 