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Special localized wavemodes show up in several physical scenarios includingBEC in optical lattices, nonlinear photonic crystals and systems with strongelectron-phonon interaction. These result from an underlying nonlinearcontribution to the wave equation that is usually assumed to be instantaneous.Here we demonstrate that the relaxation process of the nonlinearity has aprofound impact in the wavepacket dynamics and in the formation of localizedmodes. We illustrate this phenomenology by considering the one-electronwavepacket spreading in a $C60$ buckball structure whose dynamics is governedby a discrete nonlinear Schroedinger equation with a Debye relaxation of thenonlinearity. We report the full phase-diagram related to the spacial extensionof the asymptotic wavepacket and unveil a complex wavepacket dynamicalbehavior.

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