The privileged spectrum of cnoidal ion holes and its extension by imperfect ion trapping

The fundamental properties of nonlinear ion hole modes propagating in current-driven collisionless plasmas are derived. Making use of Schamel's alternative method their spatial structure ϕ(x) and phase velocities u0 are analyzed and found to depend crucially on the used trapped ion distribution fit. A regular fit represents a continuous spectrum, which is called privileged or perfect since it yields a definite u0 and appears most realistic. A singular fit, on the other hand, involving jumps and moderate slope singularities at the separatrix, does reveal further classes of hole equilibria at the cost, however, of a well-defined u0. This explains why Bernstein, Greene, Kruskal (BGK)-solutions of the Vlasov-Poisson system, exhibiting a strong slope singularity of their derived trapped particle distribution, can principally not provide definite u0s. The nonlinear dispersion relation (or u0) of privileged ion holes, on the other hand, is equivalent with that of cnoidal electron holes, i.e. in addition to the ordinary ion acoustic branch there exists a correspondence to the “Langmuir” branch and to the multiple “slow electron acoustic” branches, reflecting different trapping scenarios.