Multi-soliton interaction of nonlinear ion sound waves in a pair-ion–electron (PIE) plasma having non-Maxwellian electrons including Kappa, Cairns, and generalized two spectral index distribution functions is studied. To this end, a modified Korteweg–de Vries (mKdV) equation is obtained to investigate the ion-acoustic waves in a PIE plasma at a critical plasma composition. The effects of temperature and density ratios and the non-Maxwellian electron velocity distributions on the overtaking interaction of solitons are explored in detail. The results reveal that both hump (positive peak) and dip (negative peak) solitons can propagate for the physical model under consideration. Two and three-soliton interactions are presented, and the novel features of interacting compressive and rarefactive solitons are highlighted. The present investigation may be useful in laboratory plasmas where PIE plasmas have been reported.

The time-fractional generalized Burger–Fisher equation (TF-GBFE) has various applications across various scientific and engineering disciplines. It is used for investigating various phenomena, including the dynamics of fluid flow, gas dynamics, shock-wave formation, heat transfer, population dynamics, and diffusion transport, among other areas of research. By incorporating fractional calculus into these models, researchers can more effectively represent the non-local and memory-dependent effects frequently observed in natural phenomena. Due to the importance of the family of TF-GBFEs, this work introduces a changed iterative method for analyzing this family analytically to gain a deep understanding of many nonlinear phenomena described by this family (e.g., shock waves). The proposed approach combines two algorithms: the Laplace transform and the residual power series method. The suggested technique is thoroughly discussed. Two numerical problems are discussed to check the effectiveness and accuracy of the proposed method. The approximations for integer and fractional orders are compared with the exact solution for integer-order problems. Finally, to investigate how the fractional order affects these problems, the obtained results are discussed graphically and numerically in the tables.