The magnetosheath of Earth plays a crucial role in shaping the plasma composition of Earth’s magnetosphere. It is instrumental in controlling the thermal properties of magnetospheric plasma and is fundamental to the dynamic processes that govern the evolution of the magnetosphere. Nonlinear solitary waves are frequently observed across diverse regions of Earth’s magnetosphere, utilizing Wideband Data (WBD) plasma wave receivers aboard Cluster Wave Experiment Consortium (WEC). Previous and current space missions have enabled the meticulous observation and analysis of the characteristics of solitary waves, including their amplitude, density, temperature, and temporal durations of waves. In this study, we explore a magnetized plasma system encompassing an inertial ion fluid alongside electron populations characterized by regularized \(\kappa\) distribution, exhibiting two distinct temperature profiles (cold & hot). Such two distinct electron profiles, alongside data derived from a multitude of observations and laboratory experiments, have been meticulously integrated into the theoretical model presented in this article, aiming to enhance our comprehension of the Earth’s magnetosheath region. A nonlinear equation named Laedke–Spatschek equation is derived using perturbation framework for the plasma system. The examination focuses on propagating plane waves to explore phase-space dynamics. A magnetic field adds complexity to nonlinear wave dynamics, leading to intricate phase-space behaviour. These nonlinear waves exhibit various structures, including chaotic, quasi-periodic (irrational), multi-periodic (rational), and periodic (rational) oscillations, depending on supra-thermal indices and magnetic field strength. Additionally, our research delves into the synergistic effects that arise from the interplay between magnetic field strength, ion concentration level, and the dynamics of high-energy electron populations, all of which contribute to the overall wave dynamics. The findings from our model are anticipated to provide significant insights into not only the Earth’s magnetosheath but also broader astrophysical environments where similar conditions may occur.
