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.