# Amplitude-dependent topological edge states in nonlinear phononic lattices

## TL;DR

This paper explores how nonlinearities in phononic lattices influence topologically protected edge states, revealing amplitude-dependent frequency shifts and mode migrations, with potential for controlling wave-modes via nonlinear interactions.

## Contribution

It introduces a theoretical framework for understanding and controlling topological edge states in nonlinear phononic lattices through amplitude tuning.

## Key findings

- Localized modes arise at interfaces between inverted spring-mass chains.
- Cubic nonlinearities cause amplitude-dependent frequency shifts of localized modes.
- Numerical simulations show transition from bulk to edge modes with varying excitation amplitude.

## Abstract

This work investigates the effect of nonlinearities on topologically protected edge states in one and two-dimensional phononic lattices. We first show that localized modes arise at the interface between two spring-mass chains that are inverted copies of each other. Explicit expressions derived for the frequencies of the localized modes guide the study of the effect of cubic nonlinearities on the resonant characteristics of the interface which are shown to be described by a Duffing-like equation. Nonlinearities produce amplitude-dependent frequency shifts, which in the case of a softening nonlinearity cause the localized mode to migrate to the bulk spectrum. The case of a hexagonal lattice implementing a phononic analogue of a crystal exhibiting the quantum spin Hall effect is also investigated in the presence of weakly nonlinear cubic springs. An asymptotic analysis provides estimates of the amplitude dependence of the localized modes, while numerical simulations illustrate how the lattice response transitions from bulk-to-edge mode-dominated by varying the excitation amplitude. In contrast with the interface mode of the first example studies, this occurs both for hardening and softening springs. The results of this study provide a theoretical framework for the investigation of nonlinear effects that induce and control topologically protected wave-modes through nonlinear interactions and amplitude tuning.