The theory of motion of edges of dispersive shock waves generated after wave breaking of simple waves is developed. It is shown that this motion obeys Hamiltonian mechanics complemented by a Hopf-like equation for evolution of the background flow, which interacts with the edge wave packets or the edge solitons. A conjecture about the existence of a certain symmetry between equations for the small-amplitude and soliton edges is formulated. In the case of localized simple-wave pulses propagating through a quiescent medium, this theory provided a new approach to derivation of an asymptotic formula for the number of solitons eventually produced from such a pulse.