A small tail bump on a Maxwellian in the limit of vanishing collisions for a finite monochromatic wave

One of the more interesting features of kinetic descriptions of plasmas are resonant wave particle interactions. These often appear to be collisionless as the collision frequency does not appear in key results even though collisions are actually vital whenever the Coulomb logarithm is large. This deceptive behaviour normally occurs when a resonance is resolved by causality or by inserting a small Krook collision frequency in a linearised kinetic equation to resolve a singularity. However, the actual behaviour of the resonance is more complicated as a narrow collisional boundary layer is responsible for resolving the singular behaviour. The most remarkable example is linear Landau damping, which is normally viewed as a collisionless process. However, an evaluation by Zakharov & Karpman (1963 Sov. Phys. JETP vol. 16, pp. 351-357) proved it to be a collisional process for which collisions can never be ignored for a finite wave amplitude in the collisionless limit. This realisation implies that a small tail bump on a Maxwellian will have a growth or damping rate sensitive to collisions for a finite amplitude monochromatic plasma wave in the very weak collisionality limit. This behaviour is verified here by retaining collisions in both the island dominated limit, as well as the linear, collisional boundary layer dominated regime.