Self-steepening-induced stabilization of nonlinear edge waves at photonic valley-Hall interfaces

Localized nonlinear modes at valley-Hall interfaces in staggered photonic graphene can be described in the long-wavelength limit by a nonlinear Dirac-like model including spatial dispersion terms. It leads to a modified nonlinear Schr\"odinger equation for the wave field amplitude that remarkably incorporates a nonlinear velocity term. We show that this nonlinear velocity correction results in a counter-intuitive stabilization effect for relatively high-amplitude plane-wave-like edge states, which we confirm by calculation of complex-valued small-amplitude perturbation spectra and direct numerical simulation of propagation dynamics in staggered honeycomb waveguide lattices with on-site Kerr nonlinearity. Our findings are relevant to a variety of nonlinear photonic systems described by Dirac-like Hamiltonians.