Quantifying the effects of generalized (<i>r</i>, <i>q</i>) distribution function on modulational instability and rogue wave of electron acoustic solitons

Abstract This paper presents a comprehensive analysis of modulational instability and super freak waves in an unmagnetized collisionless plasma consisting of hot and cold electron populations as described by a generalized distribution function. We employ the reductive perturbation method together with multiple-scale perturbations to derive the nonlinear Schrödinger equation governing the evolution of electron acoustic waves. This framework facilitates an inclusive investigation of the modulational instability of the electron acoustic waves, characterized by the dispersion and nonlinear coefficients of the derived equation. Our findings reveal the existence of electron acoustic envelope solitons, facilitating a clear distinction between stable and unstable regions based on established modulational instability criteria. Notably, the analytically evaluated growth rate of modulational instability demonstrates a strong dependence on the spectral indices of the generalized distribution function and the physical parameters of the plasma system. In this regard, we employ an accurate Darboux transformation scheme to derive the first- and higher-order rational solutions positioned on a nonzero background in the modulationally unstable region. This investigation further delves the features of super freak waves with extreme amplitude excitation and examine their dependences on the spectral indices, carrier wavenumber, hot-to-cold electron densities, and the temporal coordinate of the nonlinear Schrödinger equation. The study is supposed to enhance our understanding of the properties of electron acoustic solitary waves that may occur in non-Maxwellian space plasmas.