Blow-up in the nonlinear Schroedinger dimers

What is it about?

In the AC coupler, the growth of small excitations is saturated by nonlinearity which couples the growing mode to the damped one. But why does the nonlinearity not couple the growing to decaying mode in the PT-symmetric coupler? The answer is in a different geometry of the phase space. The blowing trajectories of the PT dimer slide to infinity along a surface of a cylinder whereas their AC counterparts have to go through a narrow funnel where they are easily diverted away from their course!

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Why is it important?

The PT phase space is foliated into coaxial 2D cylinders. An implication is that the blow-up regimes are all stable.The AC phase space is genuinely 3D and not foliated. Here, all unbounded trajectories lie within a rapidly narrowing funnel. The blow-up funnel is unstable: a small perturbation is sufficient to kick a trajectory out and send it toward a stable limit cycle. Thus the difference in geometry leads to different outputs despite seemingly similar physical principles of the two devices.