Formation of Patterns in Intense Hadron Beams . The Amplitude Equation Approach ∗ Stephan

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 shown to be of Ginzburg–Landau type. Paper presented at: Second ICFA Advanced Accelerator Workshop on THE PHYSICS OF HIGH BRIGHTNESS BEAMS UCLA Faculty Center, Los Angeles November 9–12, 1999 ∗Work supported by Department of Energy contract DE–AC03–76SF00515.