Nonlinear Theory of Ion Acoustic Waves with Landau Damping

Abstract

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.

Variational approximation and the use of collective coordinates

“…A theoretical analysis based on the collective coordinate approach was carried out. Instead of using the Lagrangian formulation, Caputo and Sarels derived equations of motion for the parameters by using conservation-quantity-like integrals, which were obtained by substituting the ansatz for the solution into the PDE and averaging it after multiplications with a function (see also [27][28][29] for a similar approach). However, it is important to note that the multiplying functions were, in that study, chosen rather arbitrarily.…”

Ion acoustic traveling waves

“…In particular, Ott and Sudan (1969) developed a nonlinear theory for a nonlinear ion acoustic wave. They obtained a KdV equation, with a Cauchy Principal value integral term representing the effects of Landau damping.…”

Zero-Dissipation Limit for Nonlinear Waves

“…We illustrate the application of the zero-dissipation limit results obtained here for the equation in general form in the context of several well-known wave models. We start with the generalized KdV-Burgers equation Attention is now turned to the version of these types of wave equations originally proposed by Ott & Sudan [29] and Ostrovsky [28]. They have the form and the convergence rate is of order ν 1 2 .…”

“…Numerous numerical simulations and analytical studies have been carried out to determine the effect of such a term in these models (cf. [4,7,8,12,13,15,16,21,[27][28][29]). Laboratory studies show (1.1) with p = 1 and a suitably chosen value of ν has good predictive power in cases where nonlinear effects are not too strong (e.g.…”