Single-mode saturation of the bump-on-tail instability: Immobile ions

A one-dimensional collisionless electron plasma is considered with a velocity distribution of the ’’bump-on-tail’’ type. The ions are immobile and the eigenmodes are separated by finite periodic boundary conditions. The electron number density is increased above the neutral-stable value by a small fractional amount Δ. A single mode moves on to the positive slope region and grows until saturated at small amplitude by nonlinear effects. The saturated amplitude, frequency shift, and modified electron distribution function are determined in terms of Δ. The solution is shown to be stable and conserves energy and momentum. All singular functions appearing in higher harmonics are identified as generalized functions, and a correction to the class 1c mode adjoint is obtained. The solution of the ’’inflection-on-tail’’ problem is also given.