Wavepacket spreading dynamics under a non-instantaneous nonlinearity: Self-trapping, defocusing and focusing

Special localized wavemodes show up in several physical scenarios including BEC in optical lattices, nonlinear photonic crystals and systems with strong electron-phonon interaction. These result from an underlying nonlinear contribution to the wave equation that is usually assumed to be instantaneous. Here we demonstrate that the relaxation process of the nonlinearity has a profound impact in the wavepacket dynamics and in the formation of localized modes. We illustrate this phenomenology by considering the one-electron wavepacket spreading in a $C60$ buckball structure whose dynamics is governed by a discrete nonlinear Schroedinger equation with a Debye relaxation of the nonlinearity. We report the full phase-diagram related to the spacial extension of the asymptotic wavepacket and unveil a complex wavepacket dynamical behavior.