  We systematically construct vector solitary waves in harmonically trapped
one-dimensional two-component Bose-Einstein condensates with unequal dispersion
coefficients by a numerical continuation in chemical potentials from the
respective analytic low-density linear limits to the high-density nonlinear
Thomas-Fermi regime. The main feature of the linear states herein is that the
component with the larger quantum number has instead a smaller linear
eigenenergy, enabled by suitable unequal dispersion coefficients, leading to
new series of solutions compared with the states similarly obtained in the
equal dispersion setting. Particularly, the lowest-lying series gives the
well-known dark-anti-dark waves, and the second series yields the
dark-multi-dark states, and the following series become progressively more
complex in their wave structures. The Bogoliubov-de Gennes spectra analysis
shows that most of these states are typically unstable, but they can be
long-lived and most of them can be fully stabilized in suitable parameter
regimes.

Quantum Gases