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In the present work, we revisit two-component Bose-Einstein condensates intheir fully three-dimensional (3d) form. Motivated by earlier studies ofdark-bright solitons in the 1d case, we explore the stability of thesestructures in their fully 3d form in two variants. In one the dark soliton isplanar and trapping a planar bright (disk) soliton. In the other case, a darkspherical shell soliton creates an effective potential in which a brightspherical shell of atoms is trapped in the second component. We identify thesesolutions as numerically exact states (up to a prescribed accuracy) and performa Bogolyubov-de Gennes linearization analysis that illustrates that bothstructures can be dynamically stable in suitable intervals of sufficiently lowchemical potentials. We corroborate this finding theoretically by analyzing thestability via degenerate perturbation theory near the linear limit of thesystem. When the solitary waves are found to be unstable, we explore theirdynamical evolution via direct numerical simulations which, in turn, revealnovel waveforms that are more robust. Finally, using the SO$(2)$ symmetry ofthe model, we produce multi-dark-bright planar or shell solitons involved inpairwise oscillatory motion.

On Two-Component Dark-Bright Solitons in Three-dimensional Atomic Bose-Einstein Condensates