A Hamiltonian Perturbation Theory for the Nonlinear Vlasov Equation

The nonlinear Vlasov equation contains the full nonlinear dynamics and collective effects of a given Hamiltonian system. The linearized approximation is not valid for a variety of interesting systems, nor is it simple to extend to higher order. It is also well-known that the linearized approximation to the Vlasov equation is invalid for long times, due to its inability to correctly capture fine phase space structures. We derive a perturbation theory for the Vlasov equation based on the underlying Hamiltonian structure of the phase space evolution. We obtain an explicit perturbation series for a dressed Hamiltonian applicable to arbitrary systems whose dynamics can be described by the nonlinear Vlasov equation.