  We examine the existence and characteristics of the exotic
Fulde-Ferrel-Larkin-Ovchinnikov (FFLO) phase in a one-dimensional Fermi gas
with attractive Hubbard interactions, in the presence of spin-orbit coupling
(SOC) and Zeeman field. We show that a robust FFLO phase can be created in the
presence of attractive on-site interactions and Zeeman field, and that the
addition of SOC suppresses the FFLO order and enhances the pair formation. In
absence of SOC, the system shows four phases: Bardeen-Cooper-Schrieffer (BCS),
FFLO, multi- mode pairing and fully polarized phases by tuning the Zeeman field
h, and the quantum transition between these phases is discontinuous with
respect to h. In the presence of SOC, the transition from the BCS to FFLO phase
becomes continuous. We present a complete phase diagram of this model both in
the presence and in the absence of SOC at quarter electron filling and also
explore the effect of SOC on the FFLO phase.
