<p>The current research will analyze the Shynaray-IIA equation, a coupled model that demonstrates how induced curves evolve within nonlinear media. Utilizing the new mapping method we have found the exact form of several types of wave solutions (bright, dark, and solitary) for this coupled model. The characteristics of these solutions are illustrated through 2D and 3D graphical representations of both the real and absolute value fields associated with each solution. These illustrations illustrate clearly the physical behavior of the equation itself. In addition to soliton dynamics, the study explores multistability in a perturbed form of the system, where periodic, quasi-periodic, and chaotic responses may coexist under identical parameter settings but different initial conditions. The influence of initial constraints on the system’s long-term evolution is analyzed in detail. The obtained results provide deeper insight into the coexistence of coherent soliton structures and complex dynamical behaviors in nonlinear wave systems. The model has potential applications in nonlinear optics, optical fiber communication, and wave propagation in magneto-elastic and plasma media. The study reveals the emergence of bright, dark, kink, and singular soliton structures along with periodic, quasi-periodic, and chaotic regimes under suitable parameter settings. These findings contribute to a better theoretical understanding of nonlinear evolution equations and their practical relevance in modeling complex physical systems.</p>