# Two-Dimensional Vortex Solitons in Spin-Orbit-Coupled Dipolar Bose-Einstein Condensates
## 5 Gap Solitons in Spinor SOC BEC
### Isotropic and Anisotropic Families of the Gap Soliton  
By setting \(g=\gamma\) in Equation (1), the system now returns to a more general case with DDI, ZS and contact nonlinearity (Setting (iii)). Note that the sign of \(\beta\) can be altered by rotating the magnetic field [75], and the sign of \(g\) can be switched through Feshbach resonance. This equation actually describes a 2D spinor BEC tightly trapped in the transverse (i.e., third) direction with a trapping size \(a_{\perp}\), which requires the size of any soliton results to fulfill \(l\gg a_{\perp}\). Meanwhile, the SOC strength needs to be strong enough to manifest effects experimentally, which requires \(\beta\geq 1/a_{\perp}\). Together, these requirements give:  
\[l\beta\gg 1 \tag{13}\]  
Figure 8: Effective mass \(M\) of the SVS in spiral motion controlled by the Zeeman splitting \(\Omega\). (**a**) \(M\) versus \(\Omega\) with \(N=5\) and fixed kick \(k=0.2\). Divergence occurs at \(\Omega_{M=\infty}=0.0262\). The dashed curve is the fitting \(3/(\Omega-\Omega_{M=\infty})\). (**b**) \(\Omega_{M=\infty}\) vs. the kick strength \(\eta\) with different \(N\)[59]. and Equation (1) shows the ratio of kinetic energy and the SOC term is \(\sim 1/(l\beta)\), which implies under these settings that the kinetic term can be neglected safely. Equation (1) then reduces to:  
\[ \begin{split} i\partial_{t}\psi_{+}=\lambda\hat{D}^{[-]}\psi_{-}- \Omega\psi_{+}&-(g|\psi_{+}|^{2}+g|\psi_{-}|^{2})\psi_{+}\\ &+\beta\psi_{+}\int d\mathbf{r}^{\prime}R(\mathbf{r}-\mathbf{r}^{ \prime})(|\psi_{+}(\mathbf{r}^{\prime})|^{2}+|\psi_{-}(\mathbf{r}^{\prime})|^ {2}),\\ i\partial_{t}\psi_{-}=-\lambda\hat{D}^{[+]}\psi_{+}+\Omega\psi_{-}& -(g|\psi_{+}|^{2}+g|\psi_{-}|^{2})\psi_{-}\\ &+\beta\psi_{-}\int d\mathbf{r}^{\prime}R(\mathbf{r}-\mathbf{r}^{ \prime})(|\psi_{+}(\mathbf{r}^{\prime})|^{2}+|\psi_{-}(\mathbf{r}^{\prime})|^ {2}).\end{split}  \tag{14}\]  
Linearization to Equation (14) with \(\psi_{\pm}\sim\exp(ipx+iqy-iwt)\) leads to the dispersion relation:  
\[\omega^{2}=\Omega^{2}+\beta^{2}(p^{2}+q^{2}) \tag{15}\]  
which opens a band gap of width \(2\Omega\) in free space, as shown in Figure 9, where \(p,q\) are the quasi-momentum of the soliton in the \(x,y\) direction, respectively. Note that this energy gap will close if the kinetic terms are included [60]. Analytic and numerical solutions to Equation (14) can be carried our [60], and the results show that without DDI (\(\beta=0\)), the equation gives no soliton solutions when the contact nonlinearity is attractive (\(g<0\)) and gives the unstable Townes' solitons when contact nonlinearity is repulsive (\(g>0\)); thus, no stable solitons can be obtained under this setting without DDI. It needs to be noted that the above conclusions are drawn under conditions that the strengths of self- and cross- interactions are equal, \(g=\gamma\) in Equation (1). Actually, the relative ratio between \(g\) and \(\gamma\) can be adjusted by the Feshbach resonance technique experimentally, by applying a DC magnetic field to the spinor BEC [76; 77; 78], and very recent research revealed [79] that for unequal \(g\) and \(\gamma\), even without DDI, a continuous family of stable gap solitons exists in the upper band provided that \(\gamma/g\) is less than a certain critical value (\(\gamma/g<0.77\) in [79]). On the contrary, without contact nonlinearity (\(g=0\)), but with DDI present (\(\beta>0\)), the system can produce stable isotropic solitons near the bottom edge of the gap when the DDI takes the isotropic kernel (i.e., \(\theta=0\) in Equation (3) where the dipoles are oriented perpendicular to the system's plane) and stable anisotropic solitons near the top edge of the gap when the DDI takes the anisotropic kernel  
Figure 9: Band gap and gap solitons of spin-orbit-coupling (SOC) spinor BEC with dipole-dipole interaction (DDI), Zeeman splitting (ZS) and contact nonlinearity. (**a**) The band gap structure described by Equation (15) with \((\beta,\Omega)=(1,10)\). (**b**) Isotropic and anisotropic 2D gap solitons.