Nonlinear stationary states of the Vlasov equation in the Fourier-transformed velocity-space

We give an exact expression for the nonlinear stationary solutions of the Vlasov–Poisson equation in the Fourier-transformed velocity-space in both one and three space dimensions. We show that these solutions are entire functions of the amplitude of the electrostatic potential and that they converge to the van Kampen linear continuum eigenfunctions, if this amplitude tends to zero. We establish a correspondence between the nonlinear solutions and the BGK waves over the whole complex energy-plane. The solution corresponding to a phase space electron hole is investigated in detail.