For nonzero $\delta_0$, it is modified accordingly (see (\[eq:appB\_U\_critical\])), but the above estimate is sufficient for our qualitative description.

By the time when the system enters the fully nonlinear regime, the DW amplitude becomes $|\varphi_{\boldsymbol{k}}|\sim\varphi_0\exp{\gamma_{\boldsymbol{k}}(t_{\rm s}+t_{\rm p})}$, which can be estimated from (\[eq:Linear\_Secondary\_phi0\]) and (\[eq:ts\]) as $$|\varphi_{\boldsymbol{k}}|\sim\sqrt{\frac{q_{x}\gamma_{\boldsymbol{k}}\varphi_{\rm c}}{2\delta_{0}k_{y}^{2}}}.$$ From (\[eq:Energies\]), the corresponding DW and ZF energies are as follows: $$E_{{\rm ZF}}\sim\frac{\beta^{2}}{8(1+k_{y}^{2})^{2}},\quad
E_{{\rm DW}}\sim\frac{\beta\gamma_{\boldsymbol{k}}}{8\delta_{0}k_{y}^{2}},\label{eq:Linear_Secondary_Ec}$$ where we assumed $q_x^2\ll 1+k_y^2$. Using (\[eq:Linear\_Primary\_Omegak\]) for $\gamma_{\boldsymbol{k}}$ and assuming $D_{\boldsymbol{k}}=0$ for simplicity, we obtain $$\frac{E_{{\rm ZF}}}{E_{{\rm DW}}}\sim 1 + \frac{\delta_0^2k_y^2}{(1+k_y^2)^2}\,.$$ This shows that the ZF energy and the DW energy are roughly equal to each other when the system enters the fully nonlinear regime, since $\delta_0$ and $k_y$ are of order unity. This conclusion will be used to estimate the ZF curvature in section \[sec:DSPrediction\].

These predictions are in agreement with numerical simulations (figure \[fig:PI\_and\_SI\]). This indicates that the 4MT captures the basic dynamics of the primary and the secondary instabilities. However, as shown below, the 4MT does not capture essential features of the TI, and thus more accurate models are needed to describe the TI and the Dimits shift.

Tertiary instability {#subsec:Linear_Tertiary}
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![The first four tertiary eigenmodes found numerically using the ZF velocity 