Results from the numerical experiments. Top panel: macroscopic variables calculated from the hybrid simulations at t = 36.3τ_A. In this plot β_e = 6 × 10⁻² and β_p = 10⁻³ and k₀ = 2π/L. Bottom panel: same as the previous panel, but at a later stage, t = 80τ_A. The thick orange lines correspond to the analytical predictions for the parallel electric field, velocity (essentially J_x), and density given by Eqs. (70)–(72); the match with the numerical results is quite significant. The temporal evolution is available as an online movie.