  A spin-wave theory that includes the antisymmetric Dzyaloshinskii-Moriya
exchange interactions and long-range dipole-dipole interactions is presented
for finite-length ferromagnetic spin chains. It is found that three different
physical situations arise, depending on the direction chosen in this geometry
for the axial vector of the Dzyaloshinskii-Moriya interactions. In some cases
this leads to a tilting of the equilibrium orientations near the ends of the
chain due to interfacial effects and with consequential effects on the spectrum
of discrete dipole-exchange spin waves. When variations are introduced for the
dominant bilinear exchange interactions at the ends of the spin chain, it is
shown that localized spin waves with spatial decay characteristics may occur.
