Collisionless Damping of Nonlinear Plasma Oscillations

It is well known that the linear theory of collisionless damping breaks down after a time τ ≡ (m/eεκ)½, where κ is the wavenumber and ε is the amplitude of the electric field. Jacobi elliptic functions are now used to provide an exact solution of the Vlasov equation for the resonant electrons, and the damping coefficient is generalized to be valid for times greater than t = τ. This generalized damping coefficient reduces to Landau's result when t/τ ≪ 1; it has an oscillatory behavior when t/τ is of order unity, and it phase mixes to zero as t/τ approaches infinity. The above results are all shown to have simple physical interpretations.