A new framework for particle-wave interaction

In plasma physics, collisionless charged particles are transported following the dynamics of a meanfield Vlasov equation with a self-consistent electric field generated by the charge density. Due to the long range interaction between particles, the generating electric field oscillates and disperses like a Klein-Gordon dispersive wave, known in the physical literature as plasma oscillations or Langmuir's oscillatory waves. The oscillatory electric field then in turn drives particles. Despite its great physical importance, the question of whether such a nonlinear particle-wave interaction would remain regular globally and be damped in the large time has been an outstanding open problem. In this paper, we propose a new framework to resolve this exact nonlinear interaction. Specifically, we employ the framework to establish the large time behavior and scattering of solutions to the nonlinear Vlasov-Klein-Gordon system in the small initial data regime. The novelty of this work is to provide a detailed physical space description of particles moving in an oscillatory field and to resolve oscillations for the electric field generated by the collective interacting particles. This appears to be the first such a result analyzing oscillations in the physical phase space $\mathbb{R}^3_x\times \mathbb{R}_v^3$.