  We systematically construct a series of vector solitary waves in harmonically
trapped one-dimensional three-, four-, and five-component Bose-Einstein
condensates. These stationary states are continued in chemical potentials from
the analytically tractable low-density linear limit of respective states, as
independent linear quantum harmonic oscillator states, to the high-density
nonlinear Thomas-Fermi regime. A systematic interpolation procedure is proposed
to achieve this sequential continuation via a trajectory in the
multi-dimensional space of the chemical potentials. The Bogolyubov-de Gennes
(BdG) spectra analysis shows that all of the states considered herein can be
fully stabilized in suitable chemical potential intervals in the Thomas-Fermi
regime. Finally, we present some typical $SU(n)$-rotation-induced and
driving-induced dynamics. This method can be extended to higher dimensions and
shows significant promise for finding a wide range of solitary waves ahead.
