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Blow-up regimes in the $\mathcal{PT}$- and the $\mathcal{AC}$-dimer

In the actively coupled ($\mathcal{AC}$) pair of waveguides, the growth of small perturbations is saturated by the focussing nonlinearity that couples the linearly growing to the linearly damped mode. On the other hand, in the $\mathcal{PT}$-symmetric coupler, the focussing nonlinearity promotes the blowup of stationary light beams. The purpose of this study is to compare the nonlinear dynamics and explain the opposite effect of the same nonlinearity in the two systems. We show that while the blowup regimes are stable in the $\mathcal{PT}$-symmetric pair of waveguides, they are unstable and hence cannot be observed in the $\mathcal{AC}$-dimer.