Analysis of the dual-mode fourth-order nonlinear Schrödinger equation with parabolic law nonlinearity via two analytical algorithms

In this study, the dual-mode fourth-order nonlinear Schrödinger equation with parabolic law nonlinearity is investigated using the generalized exponential rational function method and the [Formula: see text]-expansion method. The resulting techniques yield a variety of new exact optical soliton solutions such as bright, dark, singular, mixed bright–dark, and kink-type solitons. The soliton solutions obtained are extensively validated and visualized through detailed 2D and 3D, contour diagrams, and intensity profiles. All these graphical analyses not only demonstrate the correctness of the solutions, but also provide deeper insight into their physical properties. Special focus is laid on the influence of the temporal parameters on the evolution of soliton structures, which illustrate their robustness, structural stability, and dynamic propagation properties. Insights of this kind are essential for modeling wave dynamics accurately in a dual-mode system. Furthermore, the findings make a significant contribution to the theoretical framework of nonlinear wave mechanics and have practical applications in various fields, particularly fiber optics.