Abstract
We derive nonlinear wave equations describing the propagation of slowly varying wave packets formed by topological valley-Hall edge states. We show that edge pulses break up even in the absence of spatial dispersion due to nonlinear self-steepening. Self-steepening leads to the previously unattended effect of a gradient catastrophe, which develops in a finite time determined by the ratio between the pulse's nonlinear frequency shift and the size of the topological band gap. Taking the weak spatial dispersion into account results in the formation of stable edge quasisolitons. Our findings are generic to systems governed by Dirac-like Hamiltonians and validated by numerical modeling of pulse propagation along a valley-Hall domain wall in staggered honeycomb waveguide lattices with Kerr nonlinearity.