Nonlinear instabilities driven by coherent phase-space structures

In the presence of wave dissipation, phase-space structures emerge in nonlinear Vlasov dynamics. Our theory gives a simple relation between the growth of these coherent structures and that of the wave energy. The structures can drive the wave by direct momentum exchange, which explains the existence of nonlinear instabilities in both barely unstable and linearly stable (subcritical) regimes. When dissipation is modeled by a linear term in the field equation, simple expressions of a single-hole growth rate and of the initial perturbation threshold are in agreement with numerical simulations.