Nonlinear dynamics of weakly dispersive Alfvén waves

Recent results on multi-dimensional weakly nonlinear Alfvén waves are reviewed. In the long-wave limit, the parallel propagation is governed by the Derivative Nonlinear Schrödinger (DNLS) equation which must be coupled to magnetosonic waves when dealing with non-localized solutions. This limit degenerates in the case of oblique propagation. When the dispersion is large enough (compared to the nonlinearity) to maintain circular polarization, the parallel propagation of an Alfvén-wave train is governed by the usual scalar Nonlinear Schrödinger equation (NLS). Filamentation can then occur. When the dispersion is weaker, the wave amplitude obeys a vector NLS equation with an anisotropic diffraction term and thin layers of strong gradients are formed. For oblique propagation, the diffraction vanishes with the dispersion, and growing regions of finite amplitude oscillations are formed with a typical scale intermediate between the size of the wave-packet and its wavelength.