On the equations for non-linear wave solutions of the Vlasov and Poisson equations

Butler & Gribben (1968) showed how to generalize non-linear, uniform, wave type solutions of the Vlasov-Poisson system of equations to allow for slow variations in the properties of the plasma. Here one equation of the derived set (that obtained by ensuring that the appropriate solutions of the Poisson equation are non-secular) is rearranged (i) so as to bring out the physical importance of those particles, trapped and untrapped, travelling near the wave speed, and (ii) so as to yield a form particularly suited to the case when the electric field is small.