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

Abstract

We investigate the spectral properties of one-dimensional spatially modulated nonlinear phononic lattices, and their evolution as a function of amplitude. In the linear regime, the stiffness modulations define a family of periodic and quasiperiodic lattices whose bandgaps host topological edge states localized at the boundaries of finite domains. With cubic nonlinearities, we show that edge states whose eigenvalue branch remains within the gap as amplitude increases remain localized, and therefore appear to be robust with respect to amplitude. In contrast, edge states whose corresponding branch approaches the bulk bands experience de-localization transitions. These transitions are predicted through continuation studies on the linear eigenmodes as a function of amplitude, and are confirmed by direct time domain simulations on finite lattices. Through our predictions, we also observe a series of amplitude-induced localization transitions as the bulk modes detach from the nonlinear bulk bands and become discrete breathers that are localized in one or more regions of the domain. Remarkably, the predicted transitions are independent of the size of the finite lattice, and exist for both periodic and quasiperiodic lattices. These results highlight the co-existence of topological edge states and discrete breathers in nonlinear modulated lattices. Their interplay may be exploited for amplitude-induced eigenstate transitions, for the assessment of the robustness of localized states, and as a strategy to induce discrete breathers through amplitude tuning.

Delocalization and higher-order topology in a nonlinear elastic lattice

“…A transition from edge to bulk in an elastic TI by increasing the excitation amplitude was experimentally demonstrated [39]. In addition, topological phase transition [40], self-tunability [41], edge solitons [42], discrete breathers [45], nonlinear harmonic generation [43,46], instability [44,47], and thermalization [48] were reported in nonlinear elastic TIs. However, these works are limited to the traditional first-order TIs, whilst nonlinear elastic HOTIs remain largely unexplored.…”

“…To investigate the nonlinear effect, we employ a special version of the Galerkin method-HBM-to calculate the band structure of the lattice. It has been shown that HBM is effective in studying 1D nonlinear elastic TIs [39,45], even for elastic wave propagation in strongly nonlinear periodic structures [52]. Based on HBM, we develop a method to characterize the higher-order band topology in the nonlinear regime.…”

“…Non-zero bulk polarization indicates nontrivial higher-order topology and the existence of higher-order topological states. Here the nonlinear effect on higher-order topological states in finite lattice is also explored using the HBM [45]. We consider a N-cell finite triangle Kagome lattice.…”

“…The eigenvalue µ = mω 2 of the modes of the nonlinear finite lattice is calculated and plotted in figure 6(a), and the eigenvalue of the Bloch bulk bands are also calculated and plotted, based on equation (7), as the grey areas in figure 6(a). The bulk modes of the finite lattice are almost completely within the Bloch bands, and just one mode detaches from the low-frequency Bloch band, which could develop into a localized mode [45].…”

“…However, the nonlinear effect on the observed edge mode is significant, see figure 6(d). Inspired by [45], we use the inverse participation ratio (IPR), to quantize the nonlinear effect on the localization of the mode. For the corner mode, we here use the IPR to quantize localization of corner modes, as shown in figure 6(b), where a large IPR represents a high degree of localization.…”