Appearance of inflection point modes in beam-plasma systems

In a stable unbounded Vlasov-Poisson plasma, the phase velocity of a neutrally stable mode with non-zero wavenumber must correspond to a point on the equilibrium distribution which is simultaneously a critical point and an inflection point. The appearance of such inflection point modes in the spectrum of beam-plasma systems occurs when the linear stability boundary develops a minimum at the wavenumber of the mode. In this event, the boundary of the parameter domain of linear instability coincides with the boundary of the parameter domain of rigorous (nonlinear) stability.