  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.
