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We study the longitudinal motion of beam particles under the action of asingle resonator wave induced by the beam itself. Based on the method ofmultiple scales we derive a system of coupled amplitude equations for theslowly varying part of the longitudinal distribution function and for theresonator wave envelope, corresponding to an arbitrary wave number. Theequation governing the slow evolution of the voltage envelope is show to be ofGinzburg-Landau type.Comment: LaTeX, 12 page

FORMATION OF PATTERNS IN INTENSE HADRON BEAMS: THE AMPLITUDE EQUATION APPROACH