Unidirectional wave propagation in one-dimensional first-order Hamiltonian systems

Defining the velocity of a conserved density to be the velocity of the center of gravity of this density, it is shown that for linear equations this velocity equals the weighted group velocity (with the density as weight function). For nonlinear equations, expressions for the centrovelocity of several conserved densities are derived. In particular, for a class of nonlocal equations, the centrovelocity of the energy density turns out to be some weighted average of the group velocity of the corresponding linearized equation. For a specific equation of this type, viz, the BBM equation, it is shown that upon restricting it to solutions whose initial form represents a long, low wave, the centrovelocity of the energy density is positive for all positive time.