$\mathcal{P}\mathcal{T}$-symmetry breaking in a necklace of coupled optical waveguides

We consider parity-time ($\mathcal{P}\mathcal{T}$) symmetric arrays formed by $N$ optical waveguides with gain and $N$ waveguides with loss. When the gain-loss coefficient exceeds a critical value ${\gamma }_c$, the $\mathcal{P}\mathcal{T}$-symmetry becomes spontaneously broken. We calculate ${\gamma }_c(N)$ and prove that ${\gamma }_c\to 0$ as $N\to \mathrm{\infty }$. In the symmetric phase, the periodic array is shown to support $2N$ solitons with different frequencies and polarisations.