Nonlinear Modulational Instability of an Electromagnetic Pulse in a Neutral Plasma
Using the propagation of an intense electromagnetic pulse through a neutral plasma as a model, we demonstrate how one can obtain information about the nonlinear behavior of modulated waves. Assuming no resonant instabilities, the envelope can be shown to evolve over long time scales according to a vector form of the well-known cubic nonlinear Schroedinger (NLS) equation. In the weakly relativistic regime, three distinct nonlinear effects contribute terms cubic in the amplitude and thus can be of comparable magnitude: ponderomotive forces, relativistic corrections, and harmonic generation. Modulational stability of any given system is shown to depend on polarization, frequency, composition, and these dependences are given. In the special case of a cold positron-electron plasma, the model is strictly modulationally stable for both linear and circular polarization. However the presence of an ambient magnetic field can make a decisive difference. Now modulational instability can arise within a broad range of frequencies and values of Bq, m particular for a pure positron-electron plasma. For the case of intensely propagating (relativistic) EM plane plasma waves, we find that the circularly-polarized waves are modulationally unstable under a range of conditions. We use a unified approach which illustrates how the modulational instability changes as one moves from the weakly relativistic case to the fully relativistic case. And we are able to give the first self-consistent calculation of the modulational instability properties of a slowly-modulated, fully relativistic envelope. Modulated, intensely propagating EM waves couple (in general) to longitudinal motions via the ponderomotive force. The effect of longitudinal motions is comparable to that of relativistic nonlinearities. A correct and proper expansion procedure requires solution of all the field equations up to the appropriate order, including the longitudinal equations. In particular, in the extreme relativistic limit of either an electron-positron or ion-electron plasma, waves with frequencies below twice the relativistic plasma frequency ω p , where ω p 2 ≡ 4πe 2 n 0(1/(mγ 0) + 1/(MΓ0)), are unstable. Growth occurs on a very short timescale comparable to the time for a wave packet to move past a fixed point. (This timescale is much shorter than that for spreading of the wave packet due to linear dispersion.)
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