Effects of reflected ions by the envelope of a modulated ion wave

Effects of ions reflected by the envelope of a modulated ion wave are investigated by applying the reductive perturbation method to the system of characteristic equations of the Vlasov equation. If we assume that the magnitude of the wave potential is of the order of epsilon, the ions which velocity ranges are of the order of epsilon are reflected by the envelope of the modulated wave. The effects of reflected ions appear as a precursor and it modifies the nonlinear Schroedinger equation, which governs the wave propagation due to the nonresonant ions. (auth.)