  We derive nonlinear wave equations describing the propagation of
slowly-varying wavepackets 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
then in the formation of stable edge quasi-solitons. 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.
