In the present work, we revisit two-component Bose-Einstein condensates in their fully threedimensional(3d) form. Motivated by earlier studies of dark-bright solitons in the 1d case, weexplore the stability of these structures in their fully 3d form in two variants. In one the darksoliton is planar and trapping a planar bright (disk) soliton. In the other case, a dark sphericalshell soliton creates an eective potential in which a bright spherical shell of atoms is trapped inthe second component. We identify these solutions as numerically exact states (up to a prescribedaccuracy) and perform a Bogolyubov-de Gennes linearization analysis that illustrates that bothstructures can be dynamically stable in suitable intervals of suciently low chemical potentials. Wecorroborate this nding theoretically by analyzing the stability via degenerate perturbation theorynear the linear limit of the system. When the solitary waves are found to be unstable, we exploretheir dynamical evolution via direct numerical simulations which, in turn, reveal novel waveformsthat are more robust. Finally, using the SO(2) symmetry of the model, we produce multi-dark-brightplanar or shell solitons involved in pairwise oscillatory motion.
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