Nonlinear dynamics in PT-symmetric lattices (Q2854117)

The paper addresses a discrete nonlinear Schrödinger equation for finite lattices, with the onsite self-focusing cubic nonlinearity, and mutually compensated linear gain and loss applied at alternating adjacent sites of the lattice, which corresponds to the condition of the PT-symmetry. Several rigorous results concerning the dynamics of this nonlinear non-conservative dynamical system are proved in a rigorous form. The detailed analysis is presented for the shortest lattices of this type, i.e., dimers and quadrimers. First, analyzing evolution of the norm of the solution, which is a dynamical invariant in the absence of the gain and loss, it is demonstrated that no solutions may blowup in a finite time, although exponential instability is possible. For the case of the unbroken PT symmetry, i.e., the situation in which the zero solution is neutrally stable, it is proved that solutions starting with sufficiently small initial data remain bounded at all times. On the other hand, in the same situation there are solutions which grow exponentially at large times, if the initial amplitude is large enough. The predicted results are verified by means of direct simulations of the system.