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

Topological Superfluid in one-dimensional Ultracold Atomic System with Spin-Orbit Coupling