$\mathcal{PT}$-symmetric branched optical lattices: Spectral properties and stability of solitons

Abstract We propose a model for tunable $\mathcal{PT}$-symmetric branched optical lattices by investigating both linear and nonlinear Schrödinger equations with a $\mathcal{PT}$-symmetric periodic potential on the graph and solving them by imposing weighted vertex boundary conditions. A constraint derived from these vertex conditions determines the exceptional point of the system. In the $\mathcal{PT}$ unbroken phase, this constraint enforces $\mathcal{PT}$-symmetric boundary conditions at the vertices, ensuring a purely real spectrum; its violation leads to the emergence of complex eigenvalues in the linear regime. In the nonlinear regime, the same constraint determines the linear stability of solitons: satisfying the constraint yields stable solitons, whereas violating it corresponds to unstable solitons.

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