Chaotic demonstration of the twin-core couplers with Kerr law non-linearity employing beta derivative

The generalized auxiliary equation approach is employed to derive enhanced solitary wave solutions for nonlinear directional couplers utilizing optical metamaterials. The study highlights the influence of the fractional Beta derivative parameter on soliton dynamics, demonstrating its crucial role in shaping soliton amplitudes and wave structures. Various soliton families including semi-bright solitons, solitary dark pitched solitons, single solitons, and mixed hyperbolic, trigonometric, and rational solitons are systematically constructed. Furthermore, the impact of overlapping functions on soliton interactions is investigated, revealing their significant role in amplitude modulation, wave localization, and phase shifts. This insight provides a deeper understanding of nonlinear optical interactions and enhances the accuracy of wave propagation models in directional couplers. These solutions are further analyzed using advanced computational tools to extract numerical insights. Beyond mathematical derivations, the physical relevance of these findings is explored through phase portrait analysis, quasi-periodic patterns, Lyapunov exponents, 2D Power spectrum and 3D attractors. These analyses provide a deeper understanding of energy transfer mechanisms, stability characteristics, and nonlinear optical interactions within directional couplers. The sensitivity evaluation underscores the system’s response to perturbations, offering valuable implications for the design and optimization of optical communication systems, signal processing, and photonic device engineering.