A comprehensive study of the conformable time-fractional coupled Konno–Oono equation: new methodologies and stability analysis in magnetic field

This research extensively investigates the fractional coupled Konno–Oono model, a prevalent framework in diverse scientific and engineering disciplines. The primary objective is to unravel the intricate dynamics embedded in this model. We adeptly convert partial differential equations into ordinary differential equation by applying appropriate transformations. Through the innovative approaches of the modified Khater method and the modified extended auxiliary equation method, we systematically derive solutions, encompassing periodic, dark, bright, singular, kink, and anti-kink solitons. Notably, these outcomes hold significant applications in the realm of magnetic fields. To assess the system’s stability, we employ the linear stability mechanism, offering insights into modulation instability gain. Careful selection of parametric values allows us to conduct numerical simulations, presenting results through visually engaging figures that facilitate a deeper understanding of their significance. Importantly, our findings affirm that the applied techniques not only yield complete and uniform responses but also showcase simplicity, effectiveness, and remarkable computational efficiency. It is crucial to highlight that all the results presented in this study are obtained using the Mathematica software, emphasizing the reliability and precision of our computational approach. Overall, 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.