  We propose a one-dimensional Hamiltonian $H_{1D}$ which supports Majorana
fermions when $d_{x^{2}-y^{2}}$-wave superfluid appears in the ultracold atomic
system and obtain the phase-separation diagrams both for the
time-reversal-invariant case and time-reversal-symmetry-breaking case. From the
phase-separation diagrams, we find that the single Majorana fermions exist in
the topological superfluid region, and we can reach this region by tuning the
chemical potential $\mu$ and spin-orbit coupling $\alpha_{R}$. Importantly, the
spin-orbit coupling has realized in ultracold atoms by the recent experimental
achievement of synthetic gauge field, therefore, our one-dimensional ultra-cold
atomic system described by $H_{1D}$ is a promising platform to find the
mysterious Majorana fermions.
