  The generation of magnetohydrodynamic (MHD) waves and their instabilities are
studied in galactic gaseous rotating plasmas with the effects of the magnetic
field, the self gravity, the diffusion-convection of cosmic rays as well as the
gas and cosmic-ray pressures. The coupling of the Jeans, Alfv{\'e}n and
magnetosonic waves, and the conditions of damping or instability are studied in
three different cases, namely when the propagation direction is perpendicular,
parallel and oblique to the static magnetic field, and are shown to be
significantly modified by the effects of the Coriolis force due to the rotation
of cosmic fluids and the cosmic-ray diffusion. The coupled modes can be damped
or anti-damped depending on the wave number is above or below the Jeans
critical wave number that is reduced by the effects of the Coriolis force and
the cosmic-ray pressure. It is found that the deviation of the axis of rotation
from the direction of the static magnetic field gives rise to the coupling
between the Alfv{\'e}n wave and the classical Jeans mode which otherwise
results into the modified slow and fast Alfv{\'e}n waves as well as the
modified classical Jeans modes. Furthermore, due to the effects of the cosmic
rays diffusion, there appears a new wave mode (may be called the fast Jeans
mode) in the intermediate frequency regimes of the slow and fast Alfv{\'e}n
waves, which seems to be dispersionless in the long-wavelength propagation and
has a lower growth rate of instability in the high density regimes of galaxies.
The dispersion properties and the instabilities of different kinds of MHD waves
reported here can play pivotal roles in the formation of various galactic
structures at different length scales.
