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We construct (modified) scattering operators for the Vlasov–Poisson system in three dimensions, mapping small asymptotic dynamics as    $$t\rightarrow -\infty$$  t → - ∞ to asymptotic dynamics as    $$t\rightarrow +\infty$$  t → + ∞ . 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.

Scattering Map for the Vlasov–Poisson System