Formation of Patterns and Coherent Structures in Charged Particle Beams
  In the present paper we study the long wavelength and slow time scale behavior of a coasting beam in a resonator adopting a broad-band impedance model. Based on the renormalization group approach we derive a set of coupled evolution equations for the beam envelope distribution function and the resonator voltage amplitude. The equation for the resonator voltage amplitude is further transformed into a generalized Ginzburg-Landau equation. 
Stephan I. Tzenov (Fermilab)