  The effects of linear Landau damping on the nonlinear propagation of
dust-acoustic solitary waves (DASWs) are studied in a collisionless
unmagnetized dusty plasma with two species of positive ions. The extremely
massive, micron-seized, cold and negatively charged dust particles are
described by fluid equations, whereas the two species of positive ions, namely
the cold (heavy) and hot (light) ions are described by the kinetic Vlasov
equations. Following Ott and Sudan [Phys. Fluids {\bf 12}, 2388 (1969)], and by
considering lower and higher-order perturbations, the evolution of DASWs with
Landau damping is shown to be governed by Korteweg-de Vries (KdV), modified KdV
(mKdV) or Gardner (KdV-mKdV)-like equations. The properties of the phase
velocity and the Landau damping rate of DASWs are studied for different values
of the ratios of the temperatures $(\sigma)$ and the number densities $(\mu)$
of hot and cold ions as well the cold to hot ion mass ratio $m$. The
distinctive features of the decay rates of the amplitudes of the KdV, mKdV and
Gardner solitons with a small effect of Landau damping are also studied in
different parameter regimes. It is found that the Gardner soliton points to
lower wave amplitudes than the KdV and mKdV solitons. The results may be useful
for understanding the localization of solitary pulses and associated wave
damping (collisionless) in laboratory and space plasmas (e.g., the F-ring of
Saturn) in which the number density of free electrons is much smaller than that
of ions and the heavy, micron seized dust grains are highly charged.
