profile (\[eq:Linear\_Tertiary\_sinusoidalU\]). The ordering is such that $\gamma_{\rm TI}$ decreases from left to right. The first two eigenmodes are runaway and trapped modes, respectively. The parameters are $\beta=6$, $\delta_{0}=1.5$, $q_{x}=0.4$, and $u=10$. The first row shows the eigenmode structures $\tilde{w}(x,y)=\Real[w(x){\rm e}^{{\rm i}k_{y}y}]$ ((\[eq:Linear\_Tertiary\_eigenmode\]), color), the ZF velocity $U$ (green curve), and the analytic mode structure $\tilde{w}=\Real[\mathsf{H}_m(x)\exp(S+{\rm i}k_y y)]$ ((\[eq:AnalyticTI\_S\]), dashed contour), where $m=0$ for (a1) and (b1), $m=1$ for (c1), and $m=2$ for (d1). The second row shows the corresponding Wigner function $W(x,k_x)$ ((\[eq:appA\_Wigner\]), color) and the isosurfaces of the drifton Hamiltoninan $\mathcal{H}$ ((\[eq:appB\_Drifton\_Hamiltonian\]), dashed contour). The striped structure of $W$ away from the actual location of DW quanta is a signature of a quantumlike “cat state” [@Weinbub18].[]{data-label="fig:TI_eigenvalue"}](figures/eigen){width="100.00000%"}

In the fully nonlinear regime, DW turbulence becomes inhomogeneous and localized at the extrema of the ZF velocity $U$ (figure \[fig:THhistory\]). To understand the DW dynamics in this case, let us linearize (\[eq:Equations\_mTHE\]) to obtain $$\partial_{t}\tilde{w}+U\partial_{y}\tilde{w}-(\beta+U'')\partial_{y}\tilde{\varphi}+\hat{D}\tilde{w}=0,\label{eq:Linear_Tertiary_linearDW}$$ where $$\tilde{w}=(\nabla^{2}-1+{\rm i}\hat{\delta})\tilde{\varphi},\quad U''\doteq{\rm d}^{2}U(x)/{\rm d}x^{2}.$$ For given boundary conditions in $x$, eigenmodes of (\[eq:Linear\_Tertiary\_linearDW\]) can be searched for in the form $$\tilde{w}=w(x){\rm e}^{{\rm i}(k_{y}y-\omega t)},\quad\tilde{\varphi}=\left(\frac{{\rm d^{2}}}{{\rm d}x^{2}}-k_{y}^{2}-1+{\rm i}\delta_{0}k_{y}\right)^{-1}\tilde{w},\label{eq:Linear_Tertiary_eigenmode}$$ which leads to the following equation for $w(x)$: $$\omega w=\hat{H}w,\quad \hat{H}(\hat{x},\hat{k}_x)\doteq k_{y}\hat{U}+k_{y}(\beta+\hat{U}'')\hat{\bar{k}}^{-2}-{\rm i}\hat{D},\label{eq:Linear_Tertiary_eigen}$$ where $$\hat{U}=U(\hat{x}),\quad \hat{k}_x=-{\rm i}\,{\rm 