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

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Abstract
We perform numerical investigations of the dynamical localization properties of the discrete nonlinear Schrödinger 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-André 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.