• A. Antoniazzi, D. Fanelli, S. Ruffo
  Universita di Firenze, Florence
• J. Barre'
  LANL, Los Alamos, New Mexico
• T. Dauxois
  ENS LYON, Lyon
• G. De Ninno
  ELETTRA, Basovizza, Trieste
• Y. Elskens
  Universite de Provence, Marseille

The non-linear dynamics of a single-pass, high-gain free-electron laser is analytically studied in the framework of the Vlasov-wave picture. A multiple time scale calculation is performed to derive the well known non-linear Landau equation for the complex amplitude of the unstable mode, which hence exhibits a limit cycle behaviour. The coefficients of the equation are analytically determined as function of a generic initial electron-beam distribution. Numerical results are presented. Possible extensions of this approach to the case of the Harmonic Generation are also discussed.