Scattering map for the Vlasov-Poisson system

TL;DR

This paper develops modified scattering operators for the 3D Vlasov-Poisson system, providing a new proof and leveraging Hamiltonian structure to understand long-term asymptotic behavior.

Contribution

It introduces the construction of modified wave operators and a simplified proof of modified scattering for the Vlasov-Poisson system, using Hamiltonian and pseudo-conformal techniques.

Findings

Construction of modified scattering operators for the system

A new simple proof of modified scattering

Reformulation of asymptotic analysis via local dynamics of a singular coefficient equation

Abstract

We construct (modified) scattering operators for the Vlasov-Poisson system in three dimensions, mapping small asymptotic dynamics as $t\to -\infty$ to asymptotic dynamics as $t\to +\infty$. The main novelty is the construction of modified wave operators, but we also obtain a new simple proof of modified scattering. Our analysis is guided by the Hamiltonian structure of the Vlasov-Poisson system. Via a pseudo-conformal inversion we recast the question of asymptotic behavior in terms of local in time dynamics of a new equation with singular coefficients which is approximately integrated using a generating function.