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Josephson current between two one-dimensional nanowires with proximityinduced $p$-wave superconducting pairing is calculated in the presence ofRashba spin-orbit interaction, in-plane and normal magnetic fields. We showthat Andreev retro-tunneling is realized by means of three channels. The maincontribution to the Josephson current gives a scattering in a conventionalparticle-hole channel, when an electron-like quasiparticle reflects to ahole-like quasiparticle with opposite spin yielding a current which dependsonly on the order parameters' phase differences $\varphi$ and oscillates with$4\pi$ period. Second anomalous particle-hole channel, corresponding to theAndreev reflection of an incident electron-like quasiparticle to an hole-likequasiparticle with the same spin orientation, survives only in the presence ofthe in-plane magnetic field. The contribution of this channel to the Josephsoncurrent oscillates with $4\pi$ period not only with $\varphi$ but also withorientational angle of the in-plane magnetic field $\theta$ resulting in amagneto-Josephson effect. Third anomalous particle-particle channel, whichrepresents a reflection of an electron-like (hole-like) quasiparticle to aelectron-like (hole-like) quasiparticle with opposite spin-orientation,oscillates only with the in-plane magnetic field orientation angle $\theta$. Wepresent a detailed theoretical analysis of both DC and AC Josephson effects insuch a system showing contributions from all these channels and discussexperiments which can test our theory.

Josephson current between two $p$-wave superconducting nanowires in the presence of Rashba spin-orbit interaction and Zeeman magnetic fields