  We construct rogue wave and breather solutions of a quasi-two-dimensional
Gross-Pitaevskii equation with a time-dependent interatomic interaction and
external trap. We show that the trapping potential and an arbitrary functional
parameter that present in the similarity transformation should satisfy a
constraint for the considered equation to be integrable and yield the desired
solutions. We consider two different forms of functional parameters and
investigate how the density of the rogue wave and breather profiles vary with
respect to these functional parameters. We also construct vector localized
solutions of a two coupled quasi-two-dimensional Bose-Einstein condensate
system. We then investigate how the vector localized density profiles modify in
the constant density background with respect to the functional parameters. Our
results may help to manipulate matter rogue waves experimentally in the
two-dimensional Bose-Einstein condensate systems.
