Optical solitons of time fractional Kundu–Eckhaus equation and massive Thirring system arises in quantum field theory

This study focuses on finding the soliton solutions for the time-fractional Kundu–Eckhaus equation and the time-fractional massive Thirring system using the Shehu Adomian decomposition method (SADM). The obtained solitons exhibit periodic shapes in some particular cases. In order to enhance the understanding of the physical characteristics, the presentation of 3D and contour plots involves the selection of specific parameter values for the solutions. To examine the influence of the fractional parameter \(\theta \) $\theta$ on the solutions, two-dimensional graphs are additionally provided. In order to see how the fractional parameter \(\theta \) $\theta$ impacts the solutions, 2D graphs are also presented. In order to validate and show the SADM’s proficiency, we examine the proposed method with regard to fractional order via Atangana–Baleanu in Caputo sense, Caputo, and Caputo–Fabrizio. The uniqueness and convergence of the SADM are presented. The numerical simulations are discussed both numerically and graphically. The results of this research provide perspectives into the intricate dynamics of quantum field theory and help us understand the behaviour of fractional complicated nonlinear equations and their soliton solutions.