Effect of superthermal-trapped (\(\kappa -\beta \)) electrons on the modulational instability of ion-acoustic waves

This research work focusses on examining the attributes of ion-acoustic waves (IA waves) in a plasma environment embodying superthermal-trapped electron distributions, characterised by the parameters \(\kappa \) $\kappa$ and \(\beta \) $\beta$ representing superthermality and trapping respectively. The study utilises a modified nonlinear Schrödinger equation (m-NLSE) derived using the reductive perturbation method (RPM) to model the dynamics of modulated wave packets in this plasma framework. The study investigates the impact of superthermality ( \(\kappa \) $\kappa$ ) and trapping ( \(\beta \) $\beta$ ) parameters on the envelopes of IA waves. The research explores the conditions for modulational instability (MI) of IA waves, characterised by the dispersion-nonlinearity product PQ in the m-NLSE. Bright and dark solitons are observed corresponding to different signs of PQ, i.e., \(PQ<0\) $PQ<0$ and \(PQ>0\) $PQ>0$ indicating the presence of modulationally unstable and stable regimes, respectively. The cutoff wavenumber \(k_{c}\) $k_c$ at which MI enters increases with decreasing superthermality (increasing \(\kappa \) $\kappa$ ). Interestingly, \(k_{c}\) $k_c$ is observed to be independent of the trapping parameter ( \(\beta \) $\beta$ ), suggesting that electron trapping does not significantly influence the onset of MI. Further, the effects of \(\kappa \) $\kappa$ and \(\beta \) $\beta$ on the maximum growth rate ( \(\Gamma _{\textrm{max}}\) ${\mathrm{\Gamma }}_{\text{max}}$ ) and the modulational wavenumber ( \(K_{\textrm{max}}\) $K_{\text{max}}$ ) associated with MI are also investigated.