Long-Wavelength Limit of the Two-Fluid Euler-Poisson System

Plasma is a medium filled with free electrons and positive ions. Each particle acts as a conducting fluid with a single velocity and temperature when electromagnetic fields are present. This distinction between the roles played by electrons and ions is what we refer to as the two$-$fluid description of plasma. In this paper, we investigate the dynamics of these particles in both hot and cold plasma using a collisionless ''Euler-Poisson'' system. Employing analytical and computational techniques from differential equations, we show this system is governed by the dynamics of the Korteweg$-$de Vries (KdV) equation in the long$-$wavelength limit.