## 2 Hamiltonian Structure of Periodic Nematicon Equations
The main ingredient of the global existence theory is the conservation of the Hamiltonian of the system (1), (2)  
\[H=\int_{\mathbb{R}}\left(\frac{1}{2}|u_{y}|^{2}-\frac{\alpha\beta}{4}G(|u|^{2 })|u|^{2}\right), \tag{3}\]  
and of the (optical) power \(P=\int_{\mathbb{R}}|u|^{2}\), the squared \(L^{2}-\)norm of \(u\). We use the notation \(||f||_{L^{p}}=\left(\int_{\mathbb{R}}|f|^{p}\right)^{1/p}\), \(||f||_{L^{\infty}}=\operatorname*{esssup}_{x\in\mathbb{R}}|f(x)|\). We can use these two conserved quantities to show the boundedness of some simpler quantities. By the Cauchy-Schwarz inequality we have  
\[\int_{\mathbb{R}}G(|u|^{2})|u|^{2}\leq\left(\int_{\mathbb{R}}|G(|u|^{2})|^{2} \right)^{1/2}\biggl{(}\int_{\mathbb{R}}|u|^{4}\biggr{)}^{1/2}\leq||G(|u|^{2}) ||_{L^{2}}^{2}||u||_{L^{4}}^{2}, \tag{4}\]  
and using the boundedness of \(G\) in \(L^{2}\) we obtain  
\[\int_{\mathbb{R}}G(|u|^{2})|u|^{2}\leq C^{2}||u||_{L^{4}}^{4}. \tag{5}\]  
We use that for all \(y\in\mathbb{R}\)  
\[(u(y))^{2}=\int_{-\infty}^{y}(u^{2}(s))^{\prime}ds=\int_{-\infty}^{y}2u(s)u^{ \prime}(s)ds, \tag{6}\]  
Figure 2: The periodicity of the device (or medium) in the horizontal direction (\(y-\)axis) is due to a vertical electric field applied externally. The parallel stripes represent capacitors that apply a voltage that is uniform along the beam propagation direction (\(z-\)axis), and periodic along the \(y-\)axis, see [3; 6]. The dependence on the vertical direction is not included in model (1), (2). Horizontal boundaries are also ignored. therefore  
\[|(u(y))|^{2}\leq 2\int_{-\infty}^{y}|u(s)u^{\prime}(s)|ds\leq 2\int_{\mathbb{R}}|u(s )||u^{\prime}(s)|ds\leq 2||u||_{L^{2}}||\partial_{y}u||_{L^{2}}, \tag{7}\]  
using the Cauchy-Schwarz inequality. We then have  
\[-\partial_{y}^{2}\psi+V(y)\psi+g^{2}\psi=\alpha|u(y)|^{2}, \tag{2}\]0  
and by (5) we bound the quartic part of the Hamiltonian as  
\[-\partial_{y}^{2}\psi+V(y)\psi+g^{2}\psi=\alpha|u(y)|^{2}, \tag{2}\]1  
Then  
\[-\partial_{y}^{2}\psi+V(y)\psi+g^{2}\psi=\alpha|u(y)|^{2}, \tag{2}\]2  
with \(P\) the power, a constant. The conservation of \(H\) and (10) imply that \(||u_{y}||_{L^{2}}\) must remain bounded for all \(z\in\mathbb{R}\). Also, \(\int_{\mathbb{R}}V|u|^{2}\leq||V||_{L^{\infty}}P\), thus we have a bound  
\[-\partial_{y}^{2}\psi+V(y)\psi+g^{2}\psi=\alpha|u(y)|^{2}, \tag{2}\]3  
for all \(z\in\mathbb{R}\), with \(C_{0}\) depending on \(H\) and \(P\) at \(z=0\). We now consider an equivalent discrete system using expansions in Wannier functions. We also examine some consequences of the Hamiltonian structure of system (1), (2) and of the bound (11). We start by defining the Wannier functions associated to the Schrodinger operator \(-\partial_{y}^{2}+V(y)\), with \(V\)\(b\)-periodic, see [21; 22]. Bounded solutions \(\phi_{n,k}\) (Bloch functions) and eigenvalues \(E_{n,k}\) of the periodic Schrodinger equation satisfy  
\[-\partial_{y}^{2}\psi+V(y)\psi+g^{2}\psi=\alpha|u(y)|^{2}, \tag{2}\]4  
where  
\[-\partial_{y}^{2}\psi+V(y)\psi+g^{2}\psi=\alpha|u(y)|^{2}, \tag{2}\]5  
for all \(y\in\mathbb{R}\), \(n\in\mathbb{N}\), \(k\in\mathbb{R}\). By \(V>0\) we have \(E_{n,k}>0\), furthermore  
\[-\partial_{y}^{2}\psi+V(y)\psi+g^{2}\psi=\alpha|u(y)|^{2}, \tag{2}\]6  
for all \(n\in\mathbb{N}\), \(k\), \(y\in\mathbb{R}\). Then we can consider \(k\) in any interval of length \(2\pi/b\). The index \(n\) is referred to as band index (or number). For any fixed \(k\) in an interval of length \(2\pi/b\), \(E_{n,k}\) is the \(n-th\) largest eigenvalue of (12) with boundary conditions \(\phi_{n,k}(y+b)=e^{ikb}\phi_{n,k}(y)\), implied by (13). Also, by (12), (13), the \(b-\)periodic functions \(v_{n,k}\) satisfy  
\[-\partial_{y}^{2}\psi+V(y)\psi+g^{2}\psi=\alpha|u(y)|^{2}, \tag{2}\]7  
thus for any \(k\) fixed, \(n\) labels the eigenvalues \(E_{n,k}\) in an increasing order. This equation can be also be used to compute the \(v_{n,k}\), \(E_{n,k}\) numerically for each \(k\in[0,2\pi/b)\).