Nonlinear Landau damping in a relativistic electron-ion plasma–non-local nonlinear Schrödinger-equation and Krylov Bogoliubov Mitropolsky method

Abstract

A nonlinear theory of amplitude modulation of electrostatic envelopes in an electron-ion plasma is developed taking care of relativistic effect, using the Vlasov-Poisson equation. By using the multiple scale reductive perturbation, it is shown that such evolution is characterized by the non-local, nonlinear Schrödinger equation. This non-local term is known to take care of the resonant particles having group velocity = ∂ω/∂k on a finite amplitude plasma wave. On the other hand it is well-known that waves in plasmas can undergo collision-less damping when they resonantly interact with trapped/free particles. Such collision-less damping was termed Landau damping. Our intention in this analysis is to analyze the effect of relativistic electron motion on such collision-less damping in detail. But since the corresponding effect of phase velocity resonance in this classical case is not so prominent, we have not attempted to treat it here and, as assumed, the amplitude is not very large. This gives rise to some new estimates of the frequency shift and energy transfer rate, in both relativistic and non-relativistic cases. In the next phase, we have analyzed the condition of a modulational instability in the non-local, nonlinear Schrödinger equation. Then, the Bogoliubov-Mitropolsky approach is used to study the change in the solitary wave profile due to the presence of non-local nonlinearity in the equation, which is being treated as a perturbation. Last but not the least, the magnitude of the Landau damping is also computed. In the concluding remarks we have compared the kinetic and the usual reductive perturbation approach to the nonlinear Schrödinger equation.