Averaged variational principle for autoresonant Bernstein–Greene–Kruskal modes

Whitham’s averaged variational principle is applied in studying dynamics of formation of autoresonant (continuously phase-locked) Bernstein–Greene–Kruskal (BGK) modes in a plasma driven by a chirped frequency ponderomotive wave. A flat-top electron velocity distribution is used as a model allowing a variational formulation within the water bag theory. The corresponding Lagrangian, averaged over the fast phase variable yields evolution equations for the slow field variables, allows uniform description of all stages of excitation of driven-chirped BGK modes, and predicts modulational stability of these nonlinear phase-space structures. Numerical solutions of the system of slow variational equations are in good agreement with Vlasov–Poisson simulations.