Nonlinear travelling waves in energetic particle phase space

An exact nonlinear solution is found for long-time behaviour of spontaneously formed phase space clumps/holes in dissipative plasmas with a population of energetic particles. This solution represents a Bernstein-Greene-Kruskal mode with slowly varying shape and velocity. It describes a continuous transformation of a plasma eigenmode excited just above the instability threshold into an energetic particle mode with a significantly different frequency. An electrostatic bump-on-tail instability is chosen to illustrate the analysis. However, generality of the resonant particle dynamics makes the described approach applicable to other resonance-dominated instabilities, including rapid frequency-sweeping events for Alfvenic modes in tokamaks.