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| We consider a generalized network of the discrete nonlinear Schr\"{o}dinger type. The network consists of $2N$ sites where $N$ is finite. The model is PT symmetric, i.e. it contains terms with gain and dissipation which compensate each other. The network accounts for the Kerr-type nonlinearity as well as for inter-site nonlinear coupling.
Our results can be outlined as follows.
(a) We obtain sufficient conditions of the unbroken and broken PT symmetry of the underlying linear problem.
(b) We consider the nonlinear dynamics of the dimer model ($N=1$) and show that even in the case of the unbroken PT symmetry there exist solutions unbounded in time. We also discover a new integrable configuration of the PT-symmetric dimer. All solutions of the integrable model are bounded in time provided that PT symmetry is unbroken.
(c) We focus on the existence and stability of 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 show that under certain conditions the system admits exactly $2^{N+1}-2$ stationary modes (unique up to the gauge transformation) in the large amplitude limit. |
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