Invariant solutions, lie symmetry analysis, bifurcations and nonlinear dynamics of the Kraenkel-Manna-Merle system with and without damping effect.

This work investigates the Kraenkel-Manna-Merle (KMM) system, which models the nonlinear propagation of short waves in saturated ferromagnetic materials subjected to an external magnetic field, despite the absence of electrical conductivity. The study aims to explore and derive new solitary wave solutions for this system using two distinct methodological approaches. In the first approach, the KMM system is transformed into a system of nonlinear ordinary differential equations (ODEs) via Lie group transformation. The resulting ODEs are then solved analytically using a similarity invariant approach, leading to the discovery of various types of solitary wave solutions, including bright, dark, and exponential solitons. The second approach involves applying wave and Galilean transformations to reduce the KMM system to a system of two ODEs, both with and without damping effects. This reduced system is further analyzed to investigate its bifurcation behavior, sensitivity to initial conditions, and chaotic dynamics. The analysis reveals the presence of strange multi-scroll chaotic dynamics in the presence of damping and off-boosting dynamics without damping. In addition to these approaches, the study also applies the planar dynamical theory to obtain further new soliton solutions of the KMM system. These solitons include bright, kink, dark, and periodic solutions, each of which has been visualized through 3D and 2D graphs. The results of this research provide new insights into the dynamics of the KMM system, with potential applications in magnetic data storage, magnonic devices, material science, and spintronics.