This research extensively investigates the solitary wave solutions of the fractional coupled Konno–Oono model, a prevalent framework in diverse scientific and engineering disciplines. Three mathematical methods called the simple equation method, the modified extended auxiliary equation mapping method, and the exponential-expansion method are gradually employed to derive the analytical solutions. Moreover, Mathematica 13.0 software is used to perform the analytical computations and graphical simulations. The explored outcomes have significant applications in the realm of magnetic fields. After the careful selection of parametric values under constrained conditions, some solutions are plotted in 2-dimensional and 3-dimensional spaces to understand the physical phenomena of the concerned model. Importantly, our findings affirm that the employed methods not only yield complete and uniform responses but also showcase simplicity, effectiveness, and remarkable computational efficiency. Hence, our research contributes valuable insights into the behavior of the fractional coupled Konno–Oono model, paving the way for enhanced comprehension and potential applications in magnetic field studies.

In this paper, we analyze and provide innovative soliton solutions for a (2+1)-dimensional generalized Korteweg-de Vries (gKdV) problem. We obtain phase shifts and dispersion relations by using the generalized Arnous technique and the Riccati equation approach, thus allowing different soliton solutions to be developed. Several precise solutions with special structural properties, including kink and solitary soliton solutions, are included in our study. This detailed examination demonstrates the complex behavior of the model and its capability to explain a large scale of nonlinear wave occurrences in many physical settings. Thus, in scientific domains such as fluid mechanics, plasma physics, and wave propagation in media ranging from ocean surfaces to optical fibers, our results are crucial to comprehend the principles behind the production and propagation of many complicated phenomena. Finally, we provide 2D and 3D graphs for various solutions that have been obtained using Maple.

This paper proposes a numerical method to obtain an approximation solution for the time-fractional Schrödinger Equation (TFSE) based on a combination of the cubic trigonometric B-spline collocation method and the Crank-Nicolson scheme. The fractional derivative operator is described in the Caputo sense. The $L_1-$approximation method is used for time-fractional derivative discretization. Using Von Neumann stability analysis, the proposed technique is shown to be conditionally stable. Numerical examples are solved to verify the accuracy and effectiveness of this method. The error norms $L_2$ and $L_{\infty }$ are also calculated at different values of N and t to evaluate the performance of the suggested method.

In this study, we utilize the new direct mapping method to derive optical soliton solutions for the nonlinear conformable Schrödinger equation, which accounts for group velocity dispersion coefficients and second-order spatiotemporal terms. Leveraging bifurcation and chaos theories, we conduct an in-depth analysis of the planar dynamical system associated with the equation, providing detailed graphical representations of chaotic solutions for the perturbed system. The time series of the planar system, along with a sensitivity analysis of the perturbed system, is visualized through various graphical illustrations to underscore the model’s significance and its dynamic behavior in practical applications. We derive a new family of soliton solutions including bell-shaped, dark-bright, and mixed dark-bright solitons which have not been previously reported in the literature. Further, the influence of the conformable derivative parameter and temporal dynamics on these soliton solutions is systematically investigated, highlighting the system’s importance in real-world contexts. The results highlight the potential of the proposed model in advancing optical fiber communication technologies, especially for the efficient transmission and control of ultra-fast pulses. This work contributes both theoretical novelty and practical insights into nonlinear wave propagation in advanced photonic systems.