  We examine the role of strong nonlinearity on the topologically-robust edge
state in a one-dimensional system. We consider a chain inspired from the
Su-Schrieffer-Heeger model, but with a finite-frequency edge state and the
dynamics governed by second-order differential equations. We introduce a cubic
onsite-nonlinearity and study this nonlinear effect on the edge state's
frequency and linear stability. Nonlinear continuation reveals that the edge
state loses its typical shape enforced by the chiral symmetry and becomes
generally unstable due to various types of instabilities that we analyze using
a combination of spectral stability and Krein signature analysis. This results
in an initially-excited nonlinear-edge state shedding its energy into the bulk
over a long time. However, the stability trends differ both qualitatively and
quantitatively when softening and stiffening types of nonlinearity are
considered. In the latter, we find a frequency regime where nonlinear edge
states can be linearly stable. This enables high-amplitude edge states to
remain spatially localized without shedding their energy, a feature that we
have confirmed via long-time dynamical simulations. Finally, we examine the
robustness of frequency and stability of nonlinear edge states against
disorder, and find that those are more robust under a chiral disorder compared
to a non-chiral disorder. Moreover, the frequency-regime where high-amplitude
edge states were found to be linearly stable remains intact in the presence of
small amount of disorder of both types.
