Localization of bending vibrations in the single-wall carbon nanotubes

The main result of this work is that the small-amplitude long wavelength bending oscillations of carbon nanotubes (CNT) become localized ones if the intensity of initial excitation exceeds some threshold which depends on the CNT length. This localization results from the intensive resonant interaction of zone-boundary and nearest modes in the weakly nonlinear regime that leads to loss of stability of the zone-boundary mode as to the first step. The further development of resonant interaction leads to effective confinement of energy in the part of the system only. We study this process in the terms of Limiting Phase Trajectories and demonstrate the usefulness of transition from modal to effective particles representation for description of the system under consideration. We also show that the similar tendency to localization of oscillation is the common property of the systems, the eigenvalue spectra of which are non-equidistant ones near their edges.