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The asymptotic behavior of the solutions of the second order linearized Vlasov-Poisson system around homogeneous equilibria is derived. It provides a fine description of some nonlinear and multidimensional phenomena such as the existence of Best frequencies. Numerical results for the   \begin{document}$ 1D\times1D $\end{document}   and   \begin{document}$ 2D\times2D $\end{document}   Vlasov-Poisson system illustrate the effectiveness of this approach.

Long-time behavior of second order linearized Vlasov-Poisson equations near a homogeneous equilibrium