Dynamical analysis and soliton solutions of Kraenkel-Manna-Merle system with beta time derivative

Abstract

This paper examines the fractional Kraenkel-Manna-Merle (KMM) system, which models the behavior of a nonlinear ultrashort wave pulse in non-conductive saturated ferromagnetic materials. The primary contribution of this paper is a thorough dynamical analysis of the model in non-conductive saturated ferromagnetic materials, employing the beta derivative to unveil intricate behaviors and deepen our understanding of the underlying physics. The objective is to provide a thorough analysis, including identifying solitons, studying bifurcation phenomena, exploring chaotic behavior, and assessing stability. By using the Modified Sardar subequation method, a recent addition to the literature, we uncover various soliton solutions, some of which are presented here for the first time. These solutions are visualized with 2D and 3D graphics to explore fractional effects, focusing on solitons such as bright, dark, periodic singular, kink, anti-kink, and singular kink. This method proves effective for solving a broad range of nonlinear equations in mathematical physics, offering a notable advantage in generating diverse solution families. The study also includes a detailed analysis of the model’s dynamics, covering bifurcation, chaos, and stability. Phase portrait analysis at critical points reveals the system’s transitional behavior. The addition of an external periodic force induces chaotic dynamics, shown through 2D and 3D visualizations. Stability analysis further confirms the effectiveness of these approaches in examining phase portraits and solitons across various nonlinear systems.