Fundamentals of Plasma Physics 2025, 46 - Ion Acoustic Wave Soliton

Using an acoustic wave dispersion relation

eventually coming out to the Korteweg-de-Vries equation. As the linear wave is forward traveling with unity velocity in the dim-less vars, the propagation is simply χ = χ(ξ - Vτ) = χ(η). Then, using the substitution χcosh²θ = 3V

The resulting soliton solution vanishes at |ξ| → ∞, the spatial profile consists of a solitary pulse centered around ξ = ξ₀ + V_τ, and the pulse width scales with V^1/2, so larger amplitudes are sharper, and propagate faster.