Using the Krylov–Bogoliubov–Mitropolski method, the modulational instability of ion-acoustic waves in a collisionless plasma consisting of isothermal electrons and adiabatic ions is studied. It is found that with the inclusion of ion temperature the modulational instability sets in much earlier.

Using the Krylov–Bogoliubov–Mitropolsky method of perturbation for weak non-linearities, we study the modulational instability of ion-acoustic waves in a collisionless plasma in the presence of a density gradient. We find that the density gradient of second or higher order of perturbation does not effect the dispersion and the non-linearity coefficients in the non-linear Schrödinger equation.

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The propagation of ion acoustic waves in an inhomogeneous plasma is studied in the presence of finite ion temperature. The standard nonlinear Schrödinger equation is obtained, and the regions of instability are plotted. It is claimed that these extraneous factors tend to lower the value of the critical wavenumber.

The effect of ion streaming on the modulational instability (MI) of ion-acoustic waves in a collisionless unmagnetized plasma is studied. The nonlinear Schrödinger equation is derived by using the Krylov–Bogoliubov–Mitropolsky (KBM) perturbation technique. It is found that the value of the critical wavenumber kc is highly dependent upon the ion streaming velocity V0 and the system breaks into a number of stable and unstable regions. In the weak relativistic limit of V0, instability sets in for kc <1.47. It is reconfirmed that in the absence of ion streaming the modulational instability of ion-acoustic waves sets in at a critical wavenumber kc =1.47. UFaipxr