In the present work, we focus on the cases of twosite (dimer) and three-site (trimer) configurations, i.e. oligomers, respecting the parity-time (PT ) symmetry, i.e. with a spatially odd gain-loss profile. We examine different types of solutions of such configurations with linear and nonlinear gain/loss profiles. Solutions beyond the linear PT -symmetry critical point as well as solutions with asymmetric linearization eigenvalues are found in both the nonlinear dimer and trimer. The latter feature is absent in linear PT -symmetric trimers, while both of them are absent in linear PT -symmetric dimers. Furthermore, nonlinear gain/loss terms enable the existence of both symmetric and asymmetric solution profiles (and of bifurcations between them), while only symmetric solutions are present in the linear PT -symmetric dimers and trimers. The linear stability analysis around the obtained solutions is discussed and their dynamical evolution is explored by means of direct numerical simulations. Finally, a brief discussion is also given of recent progress in the context of PT -symmetric quadrimers.
Motivated by earlier studies of artificial perceptions of light called phosphenes, we analyze traveling wave solutions in a chain of periodically forced coupled nonlinear oscillators modeling this phenomenon. We examine the discrete model problem in its co-traveling frame and systematically obtain the corresponding traveling waves in one spatial dimension. Direct numerical simulations as well as linear stability analysis are employed to reveal the parameter regions where the traveling waves are stable, and these waves are, in turn, connected to the standing waves analyzed in earlier work. We also consider a two-dimensional extension of the model and demonstrate the robust evolution and stability of planar fronts and annihilation of radial ones. Finally, we show that solutions that initially feature two symmetric fronts with bulged centers evolve in qualitative agreement with experimental observations of phosphenes.
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