TITLE
	The Landau equation for a single-pass steady-state free-electron laser: an analytical derivation

AUTHOR
	Duccio FANELLI

ABSTRACT 
	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 method allows us to obtain the well known  non-linear Landau 
	equation for the complex amplitude of the unstable mode, which hence 
	exhibits limit cycle behaviour. The coefficients of the equation are 
	analytically determined. Comparisons with numerical simulations will be also 
	discussed.