Critical Initial States in Collisionless Plasmas

We show that for collisionless plasmas with initial conditions (ICs) near “single-humped” linearly stable equilibria, there exist critical initial states that mark the transition between the ICs from which the electric field evolves to a nonzero time-asymptotic state $A(x,t)$, and those from which it Landau damps to zero. We develop an equation for $A(x,t)$ and study it as a bifurcation problem, and we obtain the asymptotic field, at leading order, as a finite superposition of waves whose frequencies obey a Vlasov dispersion relation and whose amplitudes satisfy a set of nonlinear algebraic equations.