Parametric instability of Alfv\'en wave packets

Parametric instability of Alfv\'en wave packets with monochromatic carrier wave in low-$\beta$ plasma is studied using one-dimensional magnetohydrodynamic simulations. The results show spatial growth of incoming perturbations as they propagate through the mother wave. For sufficiently short packets, the perturbations emerge downstream of the packet as small-amplitude reverse Alfv\'en waves and forward slow magnetosonic waves. For larger packets the perturbations reach non-linear amplitude while still inside the mother wave. In this case, a downstream section of the mother wave collapses but the remaining upstream section stays largely intact and enters the phase of very slow evolution. The length scale separating the linear and non-linear regimes, as well as determining the size of the surviving section in the non-linear regime, is set by the Alfv\'en crossing time of the packet, the growth rate of the parametric instability for the unmodulated carrier wave, and the amplitude of incoming perturbations. The results are discussed in connection with the physics of solar wind.