Theory of Electron Holes

We present analytical solutions for the electron holes discovered recently and for the `Gould-Trivelpiece-Soliton'. Based on appropriate electron distribution functions, stationary solutions of the Vlasov-Poisson-system adopted to finite geometry are constructed. It is shown that the electron hole found experimentally and by numerical simulations is well described by this solution. It represents a nonlinear version of the slow-electron acoustic mode and exists due to a distortion of the electron distribution in the resonant region, forming a hollow vortex in phase space. Its importance for studying nonlinear diffusion processes in phase space is emphasized.