A Lagrangian Theory for Nonlinear Wave Packets in a Collisionless Plasma

A Lagrangian, for the slowly varying complex amplitude of an almost monocbromatio electrostatio plasma wave in an unmagnetized plasma, is derived. The method is a va,riant of the averaged Lagrangian technique of whitham, adoptod for uso in plasma pbysics by employing Low's Lagrangian, together with certain elimination procedures to simplify the Lagrangian to usable form. The erpansion in powers of the wave amplitudo is ca,rried out to quartio terms, and non-adiabatic terms are also retained. Ya,riations with respect to the amplitude lead to a nonlinea,r Schriidinger equatiou for the amplitude, while variationg of the pa,rticle tnajeotories lead to a modified Vlasov equation, which inoludee the nonlinea,r reaotion of the wave on the average trajeotories. It is assumed that thuo are no rosonaJxt pa,rtioles. These equations a,ro coupled through the nonlinear fiequency shift. rt is found that the pa,rtiole aspeot of the plasma has a profound efroot on the stability of the systnm, due to resona.nce of tho wave envelope with pa,rticles moving at the group velooity. This effect, nonlinea,r Landau damping, is olways destobilizing,leading togrowbh of modulations. rn terms of tho nonlinear Scbriidinger equation, the effoct changes the equation from a lfarhetX'ook oquation with a delta-function interaction to ono with a non-momentum-conserving, non-local interaction. rn the limit of fairly small wavelength of the modulations, it is showr that the growth rato approaehes that expected from plasma kinetic theory.