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Modulation of ion sound excitations in electron–ion plasmas with double spectral index distribution function

Theoretical investigation of amplitude modulation of ion sound waves is presented here for an electron–ion plasma where the electrons are dictated by the double spectral index (r, q) distribution function. Using the standard reductive perturbative technique, a non‐linear Schrödinger (NLS) equation is derived that describes the evolution of modulated ion sound envelope excitations. Stability analysis of the NLS equation shows that the ion sound waves remain stable for the flat‐topped and kappa‐like distributions, but they can become unstable for the spiky electron velocity distribution. It is shown that changing the electron population in regions of low and high phase space density regions results in remarkable features that have no equivalent in ion sound waves with Boltzmannian electrons. Different types of localized ion sound excitations are plotted for the different shapes of the distribution functions controlled by the double spectral indices, and the underlying physics is discussed in detail. The present investigation may be beneficial to understand ion sound excitations in space plasmas where the distribution functions of the shapes presented here are frequently encountered by the satellite missions.