Non-linear dispersion of cold plasma waves

The propagation of a packet of weakly non-linear dispersive cold plasma waves is studied by means of a two-time scale expansion. The effects of amplitude dispersion and the coupling to the mean plasma motion are taken into account. The governing equations are put into a conservation form. It is found that this system of equations is elliptic or hyperbolic depending on the wave-number of the dispersive waves. In the elliptic case modulations in the wave train grow exponentially in time and a periodic wave train will be unstable in this sense. In the hyperbolic case, slow variations in the wave train propagate and the characteristic velocities give a non-linear generalization of the linear group velocity. It is shown that except for waves which have their wave vectors nearly at right angle to the unperturbed magnetic field only fast waves with wave-number less than 0.585 (Ω_i Ω_e)^½/V_α are stable; where V_a is the Alfvén velocity and Ω_ι Ω_ɛ, are the ion and electron cyclotron frequencies respectively. A process of steepening of these waves into shocks with dissipation due to wave turbulence at the head of the wave packet is suggested.