Modulation Instability of Electron Plasma Wave

A modified nonlinear Schrödinger equation obtained on the basis of the reductive perturbation theory is compared with one obtained by Dewar applying the Lagrangian method. It is shown that the both results agree each other for a special case of the one dimensional electrostatic modes in a collisionless plasma. In a case of the electron plasma wave, the nonlocal nonlinear term of the modified nonlinear Schrödinger equation leads to a modulational instability, which is reduced to the nonlinear Landau damping in the limit of small amplitude.