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Collective behavior occurs ubiquitously in nature and it plays a key role inbacterial colonies, mammalian cells or flocks of birds. Here, we examine theaverage density and velocity of self-propelled particles, which are describedby a system of partial differential equations near the flocking transition ofthe Vicsek model. This agent-based model illustrates the trend towards flockformation of animals that align their velocities to an average of those oftheir neighbors. Near the flocking transition, particle density and velocityobey partial differential equations that include a parameter $\epsilon$measuring the distance to the bifurcation point. We have obtained analyticallythe Riemann invariants in one and two spatial dimensions for the hyperbolic($\epsilon=0$) and parabolic ($\epsilon\neq 0$) system and, under periodicinitial-boundary value conditions, we show that the solutions include wavetrains. Additionally, we have found wave trains having oscillation frequenciesthat agree with those predicted by a linearization approximation and that maypropagate at angles depending on the initial condition. The wave amplitudesincrease with time for the hyperbolic system but are stabilized to finitevalues for the parabolic system. To integrate the partial differentialequations, we design a basic numerical scheme which is first order in time andspace. To mitigate numerical dissipation and ensure good resolution of the wavefeatures, we also use a high order accurate WENO5 reconstruction procedure inspace and a third order accurate Runge-Kutta scheme in time. Comparisons withdirect simulations of the Vicsek model confirm these predictions.

Bifurcation theory captures band formation in the Vicsek model of flock formation