Thresholdless discrete surface solitons and stability switchings in periodically curved waveguides

We study numerically a parametrically driven discrete nonlinear Schr\"odinger equation modelling periodically curved waveguide arrays. We show that discrete surface solitons persist, but their threshold power is altered by the drive. There are critical drives at which the threshold values vanish. We also show that parametric drives can create resonance with a phonon making a new barrier for discrete solitons. By calculating the corresponding Floquet multipliers, we find that the stability of symmetric and antisymmetric off-side discrete surface solitons switches approximately at the critical drives for thresholdless solitons.