  The propagation of weakly nonlinear dust--acoustic waves in a dusty plasma
containing two ion species with different temperatures is explored. The
nonlinear equations describing both the quadratic and cubic plasma
nonlinearities are derived. It is shown that the properties of dust--acoustic
waves depend substantially on the grain size distribution. In particular, for
solitary dust--acoustic waves with a positive potential to exist in a plasma
with distributed grain size, it is necessary that the difference between the
temperatures of two ion species be large that that in the case of unusized
grains.
