  Collective behavior occurs ubiquitously in nature and it plays a key role in
bacterial colonies, mammalian cells or flocks of birds. Here, we examine the
average density and velocity of self-propelled particles, which are described
by a system of partial differential equations near the flocking transition of
the Vicsek model. This agent-based model illustrates the trend towards flock
formation of animals that align their velocities to an average of those of
their neighbors. Near the flocking transition, particle density and velocity
obey partial differential equations that include a parameter $\epsilon$
measuring the distance to the bifurcation point. We have obtained analytically
the Riemann invariants in one and two spatial dimensions for the hyperbolic
($\epsilon=0$) and parabolic ($\epsilon\neq 0$) system and, under periodic
initial-boundary value conditions, we show that the solutions include wave
trains. Additionally, we have found wave trains having oscillation frequencies
that agree with those predicted by a linearization approximation and that may
propagate at angles depending on the initial condition. The wave amplitudes
increase with time for the hyperbolic system but are stabilized to finite
values for the parabolic system. To integrate the partial differential
equations, we design a basic numerical scheme which is first order in time and
space. To mitigate numerical dissipation and ensure good resolution of the wave
features, we also use a high order accurate WENO5 reconstruction procedure in
space and a third order accurate Runge-Kutta scheme in time. Comparisons with
direct simulations of the Vicsek model confirm these predictions.
