  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 that interacts with edge wave packets or edge solitons. A
conjecture about existence of a certain symmetry between equations for the
small-amplitude and soliton edges is formulated. In 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
produced eventually from such a pulse.
