Simple and faithful nonlinear field equations for aligning   self-propelled rods
  We derive a set of minimal yet complete 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 the fronts forming density-segregated, banded solutions, allowing us to develop a more complete analytic picture of the problem at the nonlinear level. 
Anton Peshkov, Igor S. Aranson, Eric Bertin, Hugues Chat\'e, Francesco   Ginelli
cond-mat.soft cond-mat.stat-mech