This study presents an analytical and dynamical examination of the non-linear coupled Schrödinger model, which describes non-linear wave propagation in dispersive and memory-dependent media. The model involves M-truncated and Beta derivatives. By applying the modified \(\exp (-\Upsilon (\varsigma ))\) -expansion function method, we obtained linear and rational forms of trigonometric and hyperbolic trigonometric solutions. The behavior of solutions under different parameters is further analyzed using the bifurcation technique. Additionally, the study demonstrates chaotic behavior, non-linear coupled Schrödinger model. Sensitivity analysis is also conducted to illustrate the influence of small variations in system parameters on the overall dynamics. This model yields various types of solutions, including singular complexiton, singular periodic, singular bell or singular bright, singular shape, kink, anti-kink, bright and dark soliton waves. These solutions are graphically illustrated using 2D, 3D and contour plots for suitable parameter values to reflect the physical behavior of the system. The results of this study are expected to be highly beneficial in diverse scientific fields and complex investigations.
