  General dark solitons and mixed solutions consisting of dark solitons and
breathers for the third-type Davey-Stewartson (DS-III) equation are derived by
employing the bilinear method. By introducing the two differential operators,
semi-rational solutions consisting of rogue waves, breathers and solitons are
generated. These semi-rational solutions are given in terms of determinants
whose matrix elements have simple algebraic expressions. Under suitable
parametric conditions, we derive general rogue wave solutions expressed in
terms of rational functions. It is shown that the fundamental (simplest) rogue
waves are line rogue waves. It is also shown that the multi-rogue waves
describe interactions of several fundamental rogue waves, which would generate
interesting curvy wave patterns. The higher order rogue waves originate from a
localized lump and retreat back to it. Several types of hybrid solutions
composed of rogue waves, breathers and solitons have also been illustrated.
Specifically, these semi-rational solutions have a new phenomenon: lumps form
on dark solitons and gradual separation from the dark solitons is observed.
