  It is known that the low-energy physics of the Josephson effect in the
presence of Majorana zero modes exhibits a $4\pi$ periodicity as the
Aharonov-Bohm flux varies in contrast to the $2\pi$ Josephson periodicity in
usual superconducting junctions. We study this fractional Josephson effect in
1D topological superconductors in Majorana nanowire systems by focusing on the
features of the phase-energy relations in a superconducting semiconductor
nanowire with spin-orbital coupling by including different factors operational
in experimental systems, such as short wire length, suppression of
superconducting gap, and the presence of an Andreev bound state. We show that
even in the absence of Majorana zero modes, some non-topological physical
effects can manifest a $4\pi$ periodicity of the phase-energy relation in the
Josephson junction, thus providing a false positive signal for fractional
Josephson effect with no underlying Majorana zero modes. Furthermore, we
consider several scenarios of inhomogeneous chemical potential distributions in
the superconducting nanowire leading to four Majorana bound states and
construct the effective four Majorana model to correctly describe the
low-energy theory of the Josephson effect. In this setup, multiple Majorana
zero modes can also have the $4\pi$ fractional Josephson effect, although the
underlying physics arises from Andreev bound states since two close by Majorana
bound states effectively form Andreev bound states. Our work demonstrates that
the mere observation of a fractional Josephson effect simulating $4\pi$
periodicity cannot, by itself, be taken as the definitive evidence for
topological superconductivity. This finding has important implications for the
ongoing search for non-Abelian Majorana zero modes and efforts for developing
topological qubits.
