A Hamiltonian Perturbation Theory for the Nonlinear Vlasov Equation

TL;DR

This paper develops a Hamiltonian-based perturbation theory for the nonlinear Vlasov equation, enabling more accurate analysis of complex plasma and Hamiltonian systems beyond linear approximations.

Contribution

It introduces an explicit perturbation series for a dressed Hamiltonian tailored to nonlinear Vlasov dynamics, extending the analytical tools available for such systems.

Findings

Derivation of a Hamiltonian perturbation series for the Vlasov equation

Explicit construction of a dressed Hamiltonian applicable to nonlinear systems

Improved understanding of phase space evolution in Hamiltonian systems

Abstract

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.