Finite amplitude effects on Landau damping and diminished transportation of trapped electrons

We studied the effect of finite amplitude, which was ignored in Landau's linearized theory and also studied a transition criterion between linear and nonlinear Landau damping. To adjust the finite amplitude of wave and the initial trapped electron flux J_xT0, we used particle-in-cell simulation that can uniquely modulate these parameters. We predicted on the finite amplitude of wave potential and kinetic energies; these matched the simulation exactly. Oscillation and damping rates in Landau damping agreed remarkably well with those of theory and differed substantially from theory beyond certain range of amplitude. Either large amplitude or large J_xT0 increased degree of damping; this increase diminished the transportation of trapped electrons. More accurate description of linear Landau damping coupled with the finite amplitude effect was possible. Although previous theories predict nonlinear waves in large amplitude, we did not observe the expected nonlinear wave because of diminished J_xT0 at some conditions.