Modulational stability of the ion plasma mode

The cubic Schrodinger equation satisfied by the slowly varying amplitude of the longitudinal wave in an electron-ion plasma is deduced using the method of Kakutani and Sugimoto. The analysis of the Schrodinger equation shows that the ion acoustic wave is modulationally stable and that the ion plasma oscillations are modulationally stable or unstable according to whether the wavenumber is below or above a critical value which is controlled entirely by the wave dispersion