Stationary Solitary Waves in F=1 Spin-Orbit-coupled Bose-Einstein Condensates

We consider solitary wave excitations above the ground state of $F=1$ spin-orbit coupled Bose-Einstein condensates (SOBECs). The low energy properties of SOBECs in any of the three branches of the single particle dispersion relation can be described by suitable scalar nonlinear Schr\"odinger (NLS) equations which we obtain using multiple-scale expansions. This enables us to examine a variety of different configurations, such as dark solitary waves associated with higher energy branches, as well as dark and bright structures in the lowest branch. The lowest branch can also exhibit a ``superstripe'' phase that supports solitary waves. In all cases, we provide explicit expressions for the NLS coefficients, and confirm their validity with full numerical simulations of the SOBEC system including a harmonic confining potential.