Propagation of strong electromagnetic waves in tenuous plasmas

We study the propagation of electromagnetic waves in tenuous plasmas, where the wave frequency, $\omega_0$, is much larger than the plasma frequency, $\omega_{\rm P}$. We show that in pair plasmas nonlinear effects are weak for $a_0 \ll \omega_0/\omega_{\rm P}$, where $a_0$ is the wave strength parameter. In electron-proton plasmas a more restrictive condition must be satisfied, namely either $a_0\ll 1/\omega_{\rm P}\tau_0$, where $\tau_0$ is the duration of the radiation pulse, or $a_0\ll 1$. We derive the equations that govern the evolution of the pulse in the weakly nonlinear regime. Our results have important implications for the modeling of fast radio bursts. We argue that: (i) Millisecond duration bursts with a smooth profile must be produced in a proton-free environment, where nonlinear effects are weaker. (ii) Propagation through an electron-proton plasma near the source can imprint a sub-microsecond variability on the burst profile.