2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/87588In 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.LaTeX, 9 pagesAccelerator PhysicsPattern Formation and SolitonsPlasma Physicstext