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

We study localized nonlinear modes guided by the valley-Hall domain walls in staggered photonic graphene. We describe their propagation dynamics in the long-wavelength limit by a nonlinear Dirac-like model including spatial dispersion terms. It leads to a modified nonlinear Schrödinger equation for the wave-field amplitude that remarkably incorporates a nonlinear velocity term. We find that this nonlinear velocity correction results in a counterintuitive 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 results highlight the importance of taking into account higher-order nonlinearities in topological wave media and are relevant to a variety of nonlinear photonic systems described by Dirac-like Hamiltonians.