This paper focuses on solving the precise wave solution and dynamics properties, as well as stability in a nonlinear system in space-time in terms of conformable fractional derivatives. The generalized Riccati-Bernoulli sub-ODE method along with Bäcklund transformation of analyzing the underlying model captures key nonlinear dynamics that are of interest in contemporary engineering, physics, and applied mathematics. Using a conformable fractional derivative operator we have made a leap beyond classical models in integer orders to be able to represent more realistic aspects of the effects of memory and heredity both on the complex media. We have obtained the solutions and they include new types of dark kink solitons whose propagations correspond to localized transitions between two different states of medium. The qualitative character of these solutions can be visualized by 2D profiles with different values of the fractional-order parameter ( $\alpha$) and 3D plots in the classical case. The influence of the changes in the fractional ordering on the amplitude of the waves and stability of the solitons is explained with the help of these graphical representations. Moreover, a section of the phase-space and bifurcation diagrams related to the associated planar dynamical system emphasize the sensitivity and abundance of dynamics of the model with respect to parameter perturbation. This multifaceted research enriches the knowledge about nonlinear wave propagation in multi-complex media, as well as providing insights of relevance into the dynamics and relationships between nonlinearity, dispersion, and fractional-order dynamics.