Nonlinear Theory of Ion Acoustic Waves with Landau Damping

The following macroscopic equation is shown to govern the time development of a nonlinear ion acoustic wave: ∂n∂τ + α2n ∂n∂ξ + α3 ∂3n∂ξ3 + α1(8π)1/2 P ∫ −∞+∞ ∂n∂ξ′ dξ′ξ − ξ′ = 0, where n, ξ, and τ are normalized wave amplitude, space, and time coordinates, and α1, α2, α3 are parameters which depend on the relative strengths of Landau damping, nonlinearity, and dispersion. The first three terms constitute the Korteweg-deVries equation, and the last term represents the effect of Landau damping. The equation conserves the number of particles but the wave energy can be shown to decay always. It is demonstrated that an initial waveform may either steepen or not depending on the relative size of the nonlinearity as compared to Landau damping. It is also shown that the Landau damping term causes the amplitude of a solitary wave to decay with time as (τ+τ0)−2.