Amplitude-dependent edge states and discrete breathers in non-linear modulated phononic lattices

Abstract

We explore the role of non-linearities on the spectral properties of modulated one-dimensional phononic lattices. In the linear regime, a spatial modulation of stiffness is known to produce topological gaps characterized by non-zero Chern numbers, which host topological states localized at the edges of finite domains. A continuation of the linear modes as a function of amplitude is performed, revealing a series of localization and de-localization transitions that are confirmed through direct time domain simulations. The results show that edge states whose eigenvalue branch remains within the gap remains localized, and therefore appear to be robust with respect to amplitude. In contrast, edge states whose corresponding branch approaches and remains tangential to the bulk bands experience delocalization transitions. Additionally, we observe a series of amplitude-induced phase transitions as the bulk modes become discrete breathers localized in one or more regions of the domain. Remarkably, these transitions are independent on the size of the lattice. These results bring to light the co-existence of topological edge states and discrete breathers for non-linear modulated lattices, whose interplay may exploited for amplitude-induced topological and localization transitions.