Nonlinear modes in a generalized $\mathcal{PT}$-symmetric discrete nonlinear Schrödinger equation

We generalize a finite parity-time ()-symmetric network of the discrete nonlinear Schrödinger type and obtain general results on linear stability of the zero equilibrium, on the nonlinear dynamics of the dimer model, as well as on the existence and stability of large-amplitude stationary nonlinear modes. A result of particular importance and novelty is the classification of all possible stationary modes in the limit of large amplitudes. We also discover a new integrable configuration of a -symmetric dimer.