We report numerical studies of the linear and nonlinear edge dynamics of a non-harmonically confined macroscopic fractional quantum Hall fluid. In the long-wavelength and weak excitation limit, observable consequences of the fractional transverse conductivity are recovered. The first non-universal corrections to the chiral Luttinger liquid theory are then characterized: for a weak excitation in the linear response regime, cubic corrections to the linear wave dispersion and a broadening of the dynamical structure factor of the edge excitations are identified; for stronger excitations, sizable nonlinear effects are found in the dynamics. The numerically observed features are quantitatively captured by a nonlinear chiral Luttinger liquid quantum Hamiltonian that reduces to a driven Korteweg-de Vries equation in the semiclassical limit.