  We analyze vector localized solutions of two-component Bose-Einstein
condensates (BECs) with variable nonlinearity parameter and external trap
potential through similarity transformation technique which transforms the two
coupled Gross-Pitaevskii equations into a pair of coupled nonlinear
Schr\"{o}dinger equations with constant coefficients under a specific
integrability condition. In this analysis we consider three different types of
external trap potentials: a time-independent trap, a time-dependent monotonic
trap, and a time-dependent periodic trap. We point out the existence of
different interesting localized structures, namely rogue waves, dark-and bright
soliton-rogue wave, and rogue wave-breather-like wave for the above three cases
of trap potentials. We show how the vector localized density profiles in a
constant background get deformed when we tune the strength of the trap
parameter. Further we investigate the nature of the trajectories of the
nonautonomous rogue waves. We also construct the dark-dark rogue wave solution
for repulsive-repulsive interaction of two-component BECs and analyze the
associated characteristics for the three different kinds of traps. We then
deduce single, two and three composite rogue waves for three component BECs and
discuss the correlated characteristics when we tune the strength of the trap
parameter for different trap potentials.
