Soliton Dynamics and Overlapping Phenomena of a Nonlinear Fractional Pseudo-Parabolic Model in Fluids with Stability Analysis

This article investigates the soliton dynamics and overlapping phenomena for the dominant nonlinear pseudo-parabolic physical model, specifically the time-fractional one-dimensional Oskolkov (tM-fO) model. In collaboration with the exp(-𝜑(𝜉))-expansion scheme and the improved Kudryashov’s scheme compilation, various innovative analytical solutions have been achieved in exponential, trigonometric, hyperbolic, and rational function forms. We analyze the dynamic behavior of solutions obtained from the tM-fO model for specific values of the cherished parameters, including singular, bright, and dark bell solutions, kink, and anti-kink shapes. We also investigate the comparison analysis of two solutions that overlap. The dynamics of attained waves are examined and established in 3-D, 2-D, and density plots, with detailed values of the intricate parameters being strategized. All figures are prearranged to demonstrate the properties of the innovative exact solutions. Furthermore, to describe the stable or unstable behavior of the stated model, we include a linear stability analysis.