  Motivated by the spin-momentum locking of electrons at the boundaries of
topological insulators, we study a one-dimensional system of spin-orbit coupled
massless Dirac electrons with $s$-wave superconducting pairing. As a result of
the spin-orbit coupling, our model has only two kinds of linearly dispersing
modes, which we take to be right-moving spin-up and left-moving spin-down. Both
lattice and continuum models are studied. In the lattice model, we find that a
single Majorana zero energy mode appears at each end of a finite system
provided that the $s$-wave pairing has an extended form, with the
nearest-neighbor pairing being larger than the on-site pairing. We confirm this
both numerically and analytically by calculating the winding number. Next we
study a lattice version of a model with both Schr\"odinger and Dirac-like terms
and find that the model hosts a topological transition between topologically
trivial and non-trivial phases depending on the relative strength of the
Schr\"odinger and Dirac terms. We then study a continuum system consisting of
two $s$-wave superconductors with different phases of the pairing. Remarkably,
we find that the system has a {\it single} Andreev bound state which is
localized at the junction. When the pairing phase difference crosses a multiple
of $2 \pi$, an Andreev bound state touches the top of the superconducting gap
and disappears, and a different state appears from the bottom of the gap. We
also study the AC Josephson effect in such a junction with a voltage bias that
has both a constant $V_0$ and a term which oscillates with a frequency
$\omega$. We find that, in contrast to standard Josephson junctions, Shapiro
plateaus appear when the Josephson frequency $\omega_J= 2eV_0/\hbar$ is a
rational fraction of $\omega$. We discuss experiments which can realize such
junctions.
