The long-wave, small-amplitude dynamics of obliquely propagating
Alfvén waves
is shown, using a reductive perturbative expansion, to be purely linear
not only
in one space dimension but also in the dispersionless limit in higher dimensions.
Furthermore, in the context of multidimensional wave-train modulation,
all the
diffraction coefficients are found to tend to zero with the dispersion,
while the non-linear
terms in the envelope equation remain finite. In this ‘semiclassical’
limit, the
envelope dynamics results in the formation of growing regions of finite-amplitude
oscillations with a typical scale intermediate between the size of the
wave packet
and its wavelength.