A three-dimensional kinetic nonlinear 
derivative nonlinear Schödinger equation
for Alfvén wave trains

T. Passot and P.L. Sulem
CNRS, Observatoire de la Côte d'Azur, B.P. 4229,
06304 Nice Cedex 4, France

A generalized kinetic derivative nonlinear Schrödinger (KDNLS) equation for the multidimensional dynamics of Alfvén wave trains propagating along an ambient magnetic field in a collisionless plasma is derived from the Vlasov-Maxwell equations. It retains, in addition to the Landau damping, the averaged effect along the direction of propagation of the density and pression disturbances and of the induced longitudinal hydrodynamic velocity and magnetic field. This coupling to the mean fields plays a central role in the transverse dynamics of extended wave packets, and in particular in the phenomenon of Alfvén wave filamentation. The derived system is asymptotically exact and can be used to benchmark more general Landau-fluid models for the large-scale dynamics of collisionless plasmas.