  An Enskog-like kinetic equation for self-propelled particles is solved
numerically. I study a density instability near the transition to collective
motion and find that while hydrodynamics breaks down, the kinetic approach
leads to soliton-like supersonic waves with steep leading kinks and Knudsen
numbers of order one. These waves show hysteresis, modify the transition
threshold and lead to an abrupt jump of the global order parameter if the noise
level is changed. Thus they provide a mechanism to change the second-order
character of the phase transition to first order.
