Evidence for second-order oscillations at the Best frequency in direct numerical simulations of the Vlasov equation

Quantitative evidence for second-order oscillations occurring at the frequency predicted by Best [Physica 74, 183 (1974)] and by Sedlá?ek and Nocera [J. Plasma Phys. 48, 367 (1992)] is provided by direct, ab initio numerical simulations of the Vlasov equation. These oscillations are relevant both for their intrinsic connection to the more customary plasma echo (also retrieved), for their diagnostic applications to the study of turbulence, and for the high accuracy needed to reproduce them. This latter fact is used as a sensitive test for the numerical integration, which indeed reproduces both known and new features of the oscillations, including nonlinear Landau damping, particle trapping, and the oscillations' frequencies, in excellent agreement with theory