Effects of nonlinearity on the time evolution of single-site localized states in periodic and aperiodic discrete systems.

Abstract

We perform numerical investigations of the dynamical localization properties of the discrete nonlinear Schr\"odinger equation with periodic and deterministic aperiodic on-site potentials. The time evolution of an initially single-site localized state is studied, and quantities describing different aspects of the localization are calculated. We find that for a large enough nonlinearity, the probability of finding the quasiparticle at the initial site will always be nonzero and the participation number finite for all systems under study (self-trapping). For the system with zero on-site potential, we find that the velocity of the created solitons will approach zero and their width diverge as the self-trapping transition point is approached from below. A similar transition, but for smaller nonlinearities, is found also for periodic on-site potentials and for the incommensurate Aubry-Andr\'e potential in the regime of extended states. For potentials yielding a singular continuous energy spectrum in the linear limit, self-trapping seems to appear for arbitrarily small nonlinearities. We also find that the root-mean-square width of the wave packet will increase infinitely with time for those of the studied systems which have a continuous part in their linear energy spectra, even when self-trapping has occurred.