Hole equilibria in Vlasov–Poisson systems: A challenge to wave theories of ideal plasmas

A unified description of weak hole equilibria in collisionless plasmas is given. Two approaches, relying on the potential method rather than on the Bernstein, Greene, Kruskal method and associated with electron and ion holes, respectively, are shown to be equivalent. A traveling wave solution is thereby uniquely characterized by the nonlinear dispersion relation and the “classical” potential $\mathrm{V(\phi ),}$ which determine the phase velocity and the spectral decomposition of the wave structure, respectively. A new energy expression for a hole carrying plasma is found. It is dominated by a trapped particle contribution occurring one order earlier in the expansion scheme than the leading term in conventional schemes based on a truncation of Vlasov’s equation. Linear wave theory— reconsidered by taking the infinitesimal amplitude limit—is found to be deficient, as well. Neither Landau nor van Kampen modes and their general superpositions can adequately describe these trapped particle modes due to an incorrect treatment of resonant particles for phase velocities in the thermal range. It is therefore concluded that wave theories in their present form, dictated by linearity, are not yet properly shaped to describe the dynamics of ideal plasmas (and fluids) correctly.