  We study continuations of topological edge states in the Su-Schrieffer-Heeger
model with on-site cubic (Kerr) nonlinearity, which is a 1D nonlinear photonic
topological insulator (TI). Based on the topology of the underlying spatial
dynamical system, we establish the existence of nonlinear edge states (edge
solitons) for all positive energies in the topological band gap. We discover
that these edge solitons are stable at any energy when the ratio between the
weak and strong couplings is below a critical value. Above the critical
coupling ratio, there are energy intervals where the edge solitons experience
an oscillatory instability. Though our paper focuses on a photonic system, we
also discuss the broader relevance of our methods and results to 1D nonlinear
mechanical TIs.
