  We study the effect of the intrinsic (Rashba or Dresselhaus) spin-orbit
interaction in superconductor--nanowire--superconductor (SNS) weak links in the
presence of a spin-splitting field that can result either from an intrinsic
exchange field or the Zeeman effect of an applied field. We solve the full
non-linear Usadel equations numerically and analyze the resulting supercurrent
through the weak link and the behavior of the density of states in the center
of the wire. We point out how the presence of the spin-orbit interaction gives
rise to a long-range spin triplet supercurrent, which remains finite even in
the limit of very large exchange fields. In particular, we show how rotating
the field leads to a sequence of transitions between the 0 and $\pi$ states as
a function of the angle between the exchange field and the spin-orbit field.
Simultaneously, the triplet pairing leads to a zero-energy peak in the density
of states. We proceed by solving the linearized Usadel equations, showing the
correspondence to the solutions of the full equations and detail the emergence
of the long-range supercurrent components. Our studies are relevant for
on-going investigations of supercurrent in semiconductor nanowires in the limit
of several channels and in the presence of disorder.
