  We present a detailed study of the phase diagram of the Kitaev-Hubbard chain,
that is the Kitaev chain in the presence of a nearest-neighbour density-density
interaction, using both analytical techniques as well as DMRG. In the case of a
moderate attractive interaction, the model has the same phases as the
non-interacting chain, a trivial and a topological phase. For repulsive
interactions, the phase diagram is more interesting. Apart from the previously
observed topological, incommensurate and charge density wave phases, we
identify the `excited state charge density wave' phase. In this phase, the
ground state resembles an excited state of an ordinary charge density phase,
but is lower in energy due to the frustrated nature of the model. We find that
the dynamical critical exponent takes the value $z\simeq 1.8$. Interestingly,
this phase only appears for even system sizes, and is sensitive to the chemical
potential on the edges of the chain. For the topological phase, we present an
argument that excludes the presence of a strong zero mode for a large part of
the topological phase. For the remaining region, we study the time dependence
of the edge magnetization (using the bosonic incarnation of the model). These
results further expand the region where a strong zero mode does not occur.
