We derive a set of minimal and well-behaved nonlinear field equations describing the collective properties of self-propelled rods from a simple microscopic starting point, the Vicsek model with nematic alignment. Analysis of their linear and nonlinear dynamics shows good agreement with the original microscopic model. In particular, we derive an explicit expression for density-segregated, banded solutions, allowing us to develop a more complete analytic picture of the problem at the nonlinear level.
Peshkov, A., Aranson, I. S., Bertin, E., Chate, H., & Ginelli, F. (2012). Nonlinear field equations for aligning self-propelled rods. Physical Review Letters, 109(26), . https://doi.org/10.1103/PhysRevLett.109.268701