I. INTRODUCTION For many purposes atomic Bose-Einstein condensates (BECs) can be accurately described by means of a mean-field theoretical model, the Gross-Pitaevskii (GP) equation [1–3]. Im- portantly the GP equation describes not only the ground-state, but also macroscopic non- linear excitations of BECs, such as matter-wave dark solitons, which have been studied theoretically (see, e.g., Refs. [3, 4] and references therein) and were observed in a series of experiments [5–15]. In fact, these localized nonlinear structures have attracted much at- tention as they arise spontaneously upon crossing the BEC phase-transition [16, 17], while their properties may be used as diagnostic tools for probing properties of BECs [18]. Ad- ditionally, applications of matter-wave dark solitons have been proposed: the dark soliton position can be used to monitor the phase acquired in an atomic matter-wave interferometer in the nonlinear regime [19, 20]. As matter-wave dark solitons are known to be more robust in the quasi one-dimensional (1D) geometry, the majority of relevant theoretical studies have been performed in the framework of the 1D GP equation and, particularly, in the so-called Thomas-Fermi (TF)-1D regime (see, e.g., Ref. [13]). More specifically, many works have been devoted to the stability [21–23] and dynamical properties of dark solitons, such as their oscillations [21–30] and sound emission [28, 31] in the presence of a harmonic trapping potential. In the TF-1D regime, matter-wave dark solitons have also been studied in periodic (optical lattice) potentials [32– 36], as well as in combinations of harmonic traps and optical lattices [37–40]. Notice that upon properly choosing the harmonic trap and optical lattice parameters, it is possible to create a double-well potential , as was demonstrated experimentally in Ref. [41]. It may also be formed by combining a harmonic trap with a repulsive barrier potential, induced by a blue-detuned laser beam [42]. Double-well potentials have been studied in theory [43–50], while they have played a key role in important experimental observations: these include Josephson oscillations and tunneling (for a small number of atoms) or macroscopic quantum self-trapping and an asymmetric partition of the atoms between the wells (for sufficiently large numbers of atoms) [41], as well as nonlinear matter-wave interference leading to the formation of matter-wave vortices [51, 52] and dark solitons [13, 15]. So far, there exist only a few studies on matter-wave dark solitons in double-well potentials [53, 54]. In particular, Ref. [53] was devoted to the study of nonlinearity-assisted quantum 2