Formation of Patterns in Intense Hadron Beams. The Amplitude Equation   Approach
  We study the longitudinal motion of beam particles under the action of a single resonator wave induced by the beam itself. Based on the method of multiple scales we derive a system of coupled amplitude equations for the slowly varying part of the longitudinal distribution function and for the resonator wave envelope, corresponding to an arbitrary wave number. The equation governing the slow evolution of the voltage envelope is show to be of Ginzburg-Landau type. 
Stephan I. Tzenov (SLAC)