Nonlinear Oscillations in a Collisionless Plasma

An approximate solution for electron trajectories in a space sinusoidal electric field with a slowly changing amplitude is constructed by varying the modulus of the Jacobi elliptic functions representing electron trajectories in a constant amplitude field. This solution is used, together with energy conservation, to obtain the time behavior of the amplitude of nonlinear longitudinal oscillations in a collisionless plasma. The resulting integrodifferential equation for the amplitude depends on the ratio γ/α0, where γ is the Landau damping constant, and α0 is the initial value of the frequency of oscillations of trapped electrons in the potential trough of the wave. Numerical solutions are carried out for several values of this ratio. For γ/α0 → 0 (constant amplitude limit) O'Neil's results are recovered. For small but finite values of the ratio γ/α0, the present method shows the effect of the energy exchange between resonant electrons and the decaying electric field. The time of regrowth and the correspond...