Two-dimensional dipolar gap solitons in free space with spin-orbit coupling

We present gap solitons (GSs) that can be created in free nearly two-dimensional (2D) space in dipolar spinor Bose-Einstein condensates with the spin-orbit coupling (SOC), subject to tight confinement, with size $a_{\perp }$, in the third direction. For quasi-2D patterns, with lateral sizes $l\gg a_{\perp }$, the kinetic-energy terms in the respective spinor Gross-Pitaevskii equations may be neglected in comparison with SOC. This gives rise to a bandgap in the system's spectrum, in the presence of the Zeeman splitting between the spinor components. While the present system with contact interactions does not produce 2D solitons, stable gap solitons(GSs), with vorticities $0$ and $1$ in the two components, are found, in quasi-analytical and numerical forms, under the action of dipole-dipole interaction (DDI). Namely, isotropic and anisotropic 2D GSs are obtained when the dipoles are polarized, respectively, perpendicular or parallel to the 2D plane. The GS families extend, as \textit{embedded solitons} (ESs), into spectral bands, a part of the ES branch being stable for isotropic solitons. The GSs remain stable if the competing contact interaction, with the sign opposite to that of the DDI, is included, while the addition of the contact term with the same sign destabilizes the GSs, at first replacing them by breathers, and eventually leading to destruction of the solitons. Mobility and collision of the GSs are studied too, revealing negative and positive effective masses of the isotropic and anisotropic solitons, respectively.