  Motivated by the recent finding that doped edge-sharing Cu-O chain compounds
like Na$_3$Cu$_2$O$_4$ and Na$_8$Cu$_5$O$_{10}$ are realizations of
one-dimensional Wigner crystals, we study the optical spectra of such systems.
Charge excitations in 1D Wigner crystals are described in terms of domain wall
excitations with fractional charge. We investigate analytically and numerically
the domain-wall excitations that dominate the optical absorption, and analyse
the dispersion and the parameter range of exciton states characteristic for the
long-ranged Coulomb attraction between domain walls. Here we focus on the
Wigner lattice at quarter-filling relevant for Na$_3$Cu$_2$O$_4$ and analyze in
particular the role of second neighbor hopping $t_2$ which is important in
edge-sharing chain compounds. Large $t_2$ drives an instability of the Wigner
lattice via a soft domain-wall exciton towards a charge-density wave with a
modulation period distinct from that of the Wigner lattice. Furthermore we
calculate the temperature dependence of the DC-conductivity and show that it
can be described by activated behavior combined with a $T^{-\alpha}$ dependent
mobility.
