Modulational instabilities due to trapped electrons

As a simple model for investigating the stability of an electrostatic plasma wave with trapped electrons, the plasma is regarded as a nonlinear, dispersive dielectric. A non-linear Schr\"odinger-type equation is found, which predicts instability for both electron plasma waves and ion acoustic waves. These instabilities do not depend on resonance with the bouncing motions of the trapped electrons, and thus have a different physical origin than those predicted by bunched-beam models.