Nonlinear Wave Modulation with Account of the Nonlinear Landau Damping

Abstract

The theory of nonlinear wave modulation in a collisionless plasma is developed on the basis of the Vlasov description, with particular attention to the nonlinear Landau damping process associated with resonance effects at the group velocity of the wave. Contributions from resonance particles, moving at the group velocity, change the nonlinear Schrödinger equation, derived in previous investigations, into the following equation,
(Remark: Graphics omitted.),
where φ(ξ,τ) is the small but finite potential amplitude, and τ and ξ are stretched variables in the “reductive perturbation” theory. (Remark: Graphics omitted.) denotes a Cauchy principal part. Besides giving rise to the nonlocal-nonlinear term, the contribution of these resonant particles reverse the sign of the nonlinear coupling coefficient, q, relative to the value calculated neglecting their effect.