  We address the stability and dynamics of eigenmodes in linearly-shaped
strings (dimers, trimers, tetramers, and pentamers) built of droplets of a
binary Bose-Einstein condensate (BEC). The binary BEC is composed of atoms in
two pseudo-spin states with attractive interactions, dressed by properly
arranged laser fields, which induce the (pseudo-) spin-orbit (SO) coupling. We
demonstrate that the SO-coupling terms help to create eigenmodes of particular
types in the strings. Dimer, trimer, and pentamer eigenmodes of the linear
system, which correspond to the zero eigenvalue (EV, alias chemical potential)
extend into the nonlinear ones, keeping an exact analytical form, while
tetramers do not admit such a continuation, because the respective spectrum
does not contain a zero EV. Stability areas of these modes shrink with the
increasing nonlinearity. Besides these modes, other types of nonlinear states,
which are produced by the continuation of their linear counterparts
corresponding to some nonzero EVs, are found in a numerical form (including
ones for the tetramer system). They are stable in nearly entire existence
regions in trimer and pentamer systems, but only in a very small area for the
tetramers. Similar results are also obtained, but not displayed in detail, for
hexa- and septamers.
