Existence of Periodic Waves and Compacton Solutions of the Euler–Poisson Equation: A Dynamical System Approach

This paper deals with the traveling wave solutions in the one-dimensional Euler–Poisson system equipped with the Boltzmann relation. By applying the method of dynamical systems to analyze the corresponding traveling wave system and find its phase portraits, we investigate the traveling wave solutions of the system without any restrictions. Under given parameter conditions, the existence of periodic wave solution families, compacton solution families, solitary wave solutions, kink wave solutions and anti-kink wave solutions is proved for the system with three variables. Our results complement the results in [Bae et al.; 2025] and demonstrate that the system cannot have peakon solutions.