Remarks on the parallel propagation of small-amplitude dispersive Alfven waves

The envelope formalism for the description of a small-amplitude parallel-pr opagating Alfven wave train is tested against direct numerical simulations of the Hall-MHD equations in one space dimension where kinetic effects are neglected. It turns out that the magnetosonic-wave dynamics departs from th e adiabatic approximation not only near the resonance between the speed of sound and the Alfven wave group velocity, but also when the speed of sound lies between the group and phase velocities of the Alfven wave. The modulat ional instability then does not anymore affect asymptotically large scales and strong nonlinear effects can develop even in the absence of the decay i nstability. When the Hall-MHD equations are considered in the long-waveleng th limit, the weakly nonlinear dynamics is accurately reproduced by the der ivative nonlinear Schrodinger equation on the expected time scale, provided no decay instabilities are present. The stronger nonlinear regime which de velops at later time is captured by including the coupling to the nonlinear dynamics of the magnetosonic waves.