[@Burger1999; @Becker2008; @Weller2008; @Lamporesi2013; @Anderson1999] and this has inspired a large number of theoretical works (see [@Frantzeskakis2010] and references therein). Specifically, the problem of dark solitons in BECs confined by parabolic external traps has been extensively studied [@William1997; @Muryshev1999; @Busch2000; @Pelinovsky2005].

Another interesting aspect of BECs is the study of multi-component systems, which can be described by a set of coupled GP equations. These systems were soon realized in ultracold-gas experiments by coupling two condensates made of either different atomic species [@Cornell1998] or different hyperfine states of the same atomic species [@Myatt1997]. This fact inspired the study of matter-wave solitons in these settings, and new families of solitonic structures have been found: dark-dark [@Ohberg2001; @Hoefer2011; @Yan2012], dark-bright [@Middelkamp2011; @Yan2011; @Achilleos2011], dark-antidark [@Danaila2016], etc. Particular attention has been given to the so-called Manakov limit in 1D settings, where the intra- and inter-condensate particle interactions match [@Yan2012].

The aim of this work is to study the dynamics of dark-dark solitons in a one-dimensional setting of two density-coupled condensates out of the Manakov limit, for varying coupling between condensates. Specifically, we consider a mean-field description of BECs composed by two hyperfine states of the same alkali species, and inspect both the case without 