Mixed-mode solitons in quadrupolar BECs with spin-orbit coupling

We show families of two-dimensional (2D) composite solitons in spinor quadrupolar Bose-Einstein condensates, with two localized components linearly mixed by the spin-orbit coupling (SOC), and the intrinsic nonlinearity represented by the quadrupole-quadrupole interaction (QQI) between atomic electric quadrupole moments perpendicular to the system's plane by an external electric field. Recently, stable solitons were predicted in the form of mixed mode (composites built of mixed fundamental and vortical components) in the 2D system combining the SOC and contact attractive interactions. Replacing the latter by the anisotropic long-range QQI, we demonstrate that, for a fixed norm, the system supports a continuous family of stable mixed-mode solitons (MMSs), parameterized by different norm distribution between two components. The chemical potential of the MMSs does not depend on the norm distribution, which shows a highly degeneracy. In order to identify these degenerate solitons, a magnetic field could be applied to the system. In the present system, with the Galilean invariance broken by the SOC, the composite solitons are set in motion by a kick, and we find that the motions of solitons along x and y directions are almost coupling to each other (A kick in x or y direction can probably cause a motion including x and y components). Moreover, by simulation, we also find that semi-vortex solitons (with a vortex in one component and a fundamental soliton in the other) can be supported in our model.