# Two-Dimensional Vortex Solitons in Spin-Orbit-Coupled Dipolar Bose-Einstein Condensates
## 2 The Model of SOC BEC with Dipole-Dipole Interactions  
In a two-component (spinor) BEC, a linear SOC between the two components can be induced by the combination of optical and magnetic fields [61; 62; 63]. Under mean filed approximation, the evolution of the spinor BEC wavefunction, \(\psi=(\psi_{+},\psi_{-})\), in a 2D setting is governed by the coupled Gross-Pitaevskii equation, which can be written in a normalized full form:  
\[\begin{split} i\partial_{t}\psi_{+}&=-\frac{1}{2} \
abla^{2}\psi_{+}+\lambda\hat{D}^{[-]}\psi_{-}-\Omega\psi_{+}&-(g| \psi_{+}|^{2}+\gamma|\psi_{-}|^{2})\psi_{+}\\ &+\beta\psi_{+}\int d\mathbf{r}^{\prime}R(\mathbf{r}-\mathbf{r}^{ \prime})(|\psi_{+}(\mathbf{r}^{\prime})|^{2}+|\psi_{-}(\mathbf{r}^{\prime})|^{ 2}),\\ i\partial_{t}\psi_{-}&=-\frac{1}{2}\
abla^{2}\psi_{-} -\lambda\hat{D}^{[+]}\psi_{+}+\Omega\psi_{-}&-(\gamma|\psi_{+}|^{ 2}+g|\psi_{-}|^{2})\psi_{-}\\ &+\beta\psi_{-}\int d\mathbf{r}^{\prime}R(\mathbf{r}-\mathbf{r}^{ \prime})(|\psi_{+}(\mathbf{r}^{\prime})|^{2}+|\psi_{-}(\mathbf{r}^{\prime})|^{ 2}).\end{split} \tag{1}\]  
In the equation, \(\hat{D}^{[\pm]}\) is the anisotropic Rashba SO-coupling term:  
\[\hat{D}^{[\mp]}=\lambda_{x}\partial_{x}\pm i\lambda_{y}\partial_{y}, \tag{2}\]  
where \(\lambda_{x,y}\in[0,1]\) is the strength of SO-coupling in the \(x\) and \(y\) direction, respectively. \(\Omega\) represents the strength of Zeeman splitting between the two components, which may be replaced by the Stark-Lo Surdo splitting induced by the DC electric field. \(g\) and \(\gamma\) are coefficients of self- and cross-interactions of the components. \(\beta\) is the strength of the DDI interactions; for simplicity, only density-density interactions are considered; thus, the exchange and relaxation term between different spin components has been neglected [38] (the full DDI interaction term can be seen in [62]). What is more, as the following results will show that contact nonlinearity is not a dominant factor for the physics studied here, for convenience, the intra-component contact interactions are taken to be equal. \(R(\mathbf{r}-\mathbf{r}^{\prime})\) is the DDI's kernel, which reads:  
\[R(\mathbf{r}-\mathbf{r}^{\prime})=\frac{1-3\cos^{2}\theta}{(\epsilon^{2}+| \mathbf{r}-\mathbf{r}^{\prime}|^{2})^{3/2}} \tag{3}\]  
where cutoff \(\epsilon\) is determined by the confinement in the transverse (third) dimension [30; 64; 65; 30], all the dipoles are polarized to the same direction and \(\theta\) is the angle between the polarization direction and \((\mathbf{r}-\mathbf{r}^{\prime})\). Note that the sign of \(\Omega,g,\gamma,\beta\) can all be switched between positive and negative experimentally [58; 59; 60].  
Depending on the specific settings addressed, Equation (1) reduces to different forms, which are listed briefly as follows and investigated in detail in the following sections.  
1. For \(\Omega=g=\gamma=0\), \(\beta>0\), the system corresponds to a spinor SOC BEC with pure DDI.
2. For \(g=\gamma=0\), \(\Omega\
eq 0\), \(\beta>0\), the system corresponds to a spinor SOC BEC with DDI and ZS.
3. For \(g,\gamma\
eq 0\), \(\Omega\
eq 0\), \(\beta>0\), the system corresponds to a spinor SOC BEC with DDI, ZS and contact nonlinearity.