Two-dimensional anisotropic vortex–bright soliton and its dynamics in dipolar Bose–Einstein condensates in optical lattice

We study construction and dynamics of two-dimensional (2D) anisotropic vortex–bright (VB) soliton in spinor dipolar Bose–Einstein condensates confined in a 2D optical lattice (OL), with two localized components linearly mixed by the spin–orbit coupling and long-range dipole–dipole interaction (DDI). It is found that the OL and DDI can support stable anisotropic VB soliton in the present setting for arbitrarily small value of norm N. We then present a new method via examining the mean square error of norm share of bright component to implement stability analysis. It is revealed that one can control the stability of anisotropic VB soliton only by adjusting OL depth for a fixed DDI. In addition, the dynamics of the anisotropic VB soliton was studied by applying the kick to them. The mobility of the single kicked VB soliton is Rabbi-like oscillation. However, for the collision dynamics of two kicked anisotropic VB solitons, their properties mainly depend on their initial distance and OL, and they can realize the transition from the bright component to the vortex component. Our work may provide a convenient way to prepare and manipulate anisotropic VB soliton in high-dimensional space.

Recently, the vortex-bright (VB) soliton as a twocomponent counterpart of the vortex structure in the context of multicomponent BECs [38][39][40][41][42][43][44][45] has been studied, where a bright soliton of one component is trapped by the vortex core of the other component despite that a bright soliton cannot exist on its own in a one-component repulsive condensate. VB soliton can be implemented using a spatially dependent spin interconversion and phase imprinting in spinor condensates [19]. Naturally, stabilizing VB soliton is important, and previous works have already paid much attention to confine VB solitons with external potentials [39,41], particularly optical lattice (OL) [38]. It was recently demonstrated that the means for the creation of semivortex-bright soliton in 2D is provided by using a two-component BEC with the linear spin-orbit coupling (SOC) between the components [46][47][48], which relies on the use of Rashba SOC and local self-attractive interaction. Another possibility for the creation of 2D anisotropic VB soliton is offered by using combination of SOC and long-range dipole-dipole interaction (DDI), which requires the norm exceeds a critical value; otherwise, the VB soliton is subject to a weak oscillatory instability [49,50]. Naturally, an important question arises: Can the stable anisotropic VB soliton appear in a system with arbitrary norm value? In particular, the study on the collision dynamics of two anisotropic VB solitons is still lacking.
In this paper, we address above these important questions by investigating stability and dynamics of 2D anisotropic VB soliton in the model of the spinor BEC with the SOC and DDI confined in a 2D OL. It is found that the OL and DDI can support stable anisotropic VB soliton in the present setting for arbitrarily small norm value, which is different from the case without OL in Ref. [50]. Then we present a method via examining the mean square error of norm share of bright component to implement stability analysis. It is revealed that one can control the stability of anisotropic VB soliton only by adjusting OL depth for a fixed DDI strength. In addition, the dynamics of the anisotropic VB soliton was studied by applying the kick to them. The mobility of the single kicked anisotropic VB soliton is Rabbilike oscillation. However, for the collision dynamics of two kicked anisotropic VB solitons, their properties mainly depend on their initial distance (x 0 , y 0 ) and OL. For the case in free space, it has been demonstrated that two anisotropic VB solitons with small initial distance may merge after inelastically colliding and they do not collide each other when they have big initial distance. For the case in the presence of OL, they always merge after inelastically colliding and can induce the transition from the bright component to the vortex component depending on the initial distance (x 0 , y 0 ).
The rest of the paper is structured as follows. In Sect. 2, we give a physical description of the model. In Sect. 3, we discuss the influence of OL and DDI on static properties of anisotropic VB soliton. In this section, we show the basic properties of anisotropic VB soliton in the first subsection and then implement stability analysis in the second subsection. In Sect. 4, we study the dynamics of anisotropic VB soliton by applying the kick to them. At last, we give a conclusion in Sect. 5.

Model
The system of 2D Gross-Pitaevskii equations (GPEs) for the spinor wave function, ψ = (ψ 1 , ψ 2 ), of the binary BEC with DDI and the SOC of the Rashba type, with strength λ, is written in the scaled form as with the confining potential [38] V (x, y) = 1 2 2 (x 2 + y 2 ) The two terms of potential, respectively, represent the parabolic magnetic trap and the OL, where is the strength of parabolic magnetic potential, D is the lattice constant and V 0 is the lattice depth. The strength of the DDI R r − r is where r ( ) ≡ (x ( ) , y ( ) ) is the vector of the direction and is a regularization scale, which is determined by the transverse size of the nearly-2D layer [50,51]. The form of this DDI implies that the dipoles are polarized by an external magnetic field in the 2D plane along the positive direction of the x axis; hence, In this connection, it is worth mentioning that the DDI, induced by external magnetic field, and the OL trap may be manipulated in a completely independent fashion. Therefore, to explore the effect of OL and DDI on the stability and dynamics of anisotropic VB soliton, we will choose the fixed values λ = 1, = 0.5 and D = 1.28 throughout this paper.
The linear SOC and nonlinear DDI terms can introduce the spatial anisotropy of the system. Therefore, stationary solutions of Eq. (1) are introduced as usual with total norm and energy Here μ is chemical potential and E K in,O L,SOC,D D I are densities of kinetic, OL, SOC and DDI energies, respectively. The prototypical stationary solution of Eq. (1) that will be the building block for our considerations is the single anisotropic VB soliton state. It is expected that there exist stationary anisotropic VB soliton at the trap center, by means of the imaginary time integration method, and the ansatz where A 1,2 and α 1,2 > 0 are real constants. Numerically generated results demonstrate that the stable anisotropic VB soliton can arise for arbitrarily small values of N , which is different from the case without optical lattice in Ref. [50]. A typical example of the output in the form of stable anisotropic VB soliton with norm N = 0.2 is displayed in Fig. 1 a-d.
For such a state, both components feature a typical anisotropic structure due to DDI, being stretched along the horizontal axis, and the density patterns in both components are symmetric with respect to the x and y axes. In addition, we also inspect the total energy and chemical potential of anisotropic VB soliton versus the total norm N , as shown in Fig. 1e and f. Obviously, the total energy and the chemical potential decrease with N . Hence, the anisotropic VB soliton also satisfies the famous Vakhitov-Kolokolov criterion, dμ/d N < 0, which is a necessary stability condition for any soliton supported by attractive interactions [52,53]. This means that the DDI energy plays a domain role at large N . In particular, compared with the case without optical lattice [50], there does not exist a area that is not detected by the Vakhitov-Kolokolov condition, which means anisotropic VB soliton can be stabilized by OL.

Basic properties of anisotropic VB soliton
In order to study the quantify properties of anisotropic VB soliton influenced by OL and DDI in our system, we define the following characteristics: (i) effective horizontal (effx) and vertical (effy) sizes of the bright, "1 (φ 1 ), and vortex, "2 (φ 2 ), components, where the corresponding coordinates of the center of mass X 1,2 mc and Y 1,2 mc are defined as (ii) the anisotropy of each component, and (iii) norm shares of the bright and vortex components: These characteristics as well as the total energy of the anisotropic VB soliton are displayed versus V 0 and in Fig. 2. Figure 2a and b demonstrates that the anisotropy degree η 1,2 decreases with the increase of , initially increases to a peak value and then decreases to a constant value with the increase of V 0 for the bigger value of ; hence, there forms a peak at V 0 ≈ 12. Moreover, the anisotropy of the bright component is greater than that of the vortex component for the same parameter. For a strong strength of DDI ( < 0.07), the DDI energy plays a domain role; hence, the anisotropy degrees of both two components hardly affected by the optical lattice. However, for a weak strength of DDI ( ≥ 0.07), the introduction of shallow OL can drag the anisotropic VB soliton structure; hence, its size in the x direction may be larger than that along y, and the anisotropy degree initially increases with V 0 , for V 0 ≤ 12. The OL begins to bind the anisotropic VB soliton along with the increase in the depth of OL; hence, the anisotropy degree decreases with the increase of V 0 .
Further, Fig. 2c shows that the norm share of bright component decreases with the decrease of , and increases with the growth of V 0 . Interestingly, the competition between the DDI strength and the OL depth on the norm share of bright component may provide a method to implement stability analysis of anisotropic VB soliton, which will be demonstrated in detail in the next subsection. We also discuss the total energy, and each component of E varies with V 0 and in Fig. 2d-h. We can see E K in and E SOC decrease with the decrease of , and increase with the growth of V 0 , E O L increases with the growth of V 0 and E D D I increases with the growth of . The order of magnitude of the variation

Stability analysis of anisotropic VB soliton
Based on above stationary solutions, we now discuss the stability of anisotropic VB soliton. Generally, dynamical method is used as a criterion of stability analysis for the system with anisotropic DDI. Here, we will present a new method via examining the mean square error of norm share of bright component. To do this, we define the mean square error of norm share of bright component as where the time-dependent norm share of the bright component F 1 (t) = |ψ 1 (x, y, t)| 2 dxdy/N , the average value of the norm share of the bright component in the evolution F 1 = F 1 (t) and the variance F 1 (t) = F 1 (t) − F 1 . If the mean square error of F 1 is less than a very small critical value, we believe the anisotropic VB soliton is stable; otherwise, it is unstable. It is worth noting that the mean square errors of F 1 and F 2 are equivalent [see appendix A]. To verify the above results, we give the parameter regions of stability and instability for anisotropic VB soliton in Fig. 3a, by setting critical value ( 2 F 1 (t) = 1 × 10 −6 ). The instable region in the upper left corner is due to the fact that the weak OL depth and DDI strength do not bind the anisotropic VB soliton, while the instable region in the lower right corner is due to the fact that the OL depth and DDI strength are too large, and then, the anisotropic VB soliton is not quasi-2D object and can excites arbitrarily in space. Further, we also give dynamical evolution for the system being in instable and stable parameter regions in Fig. 3b-e. As expected, two components develop an oscillatory instability in instable parameter region, and they are stable in stable parameter region. In particular, one can control the stability of anisotropic VB soliton only by adjusting OL depth V 0 for a fixed DDI strength.

The anisotropic VB soliton dynamics
As the SOC terms break the Galilean invariance of the underlying Eq. (1), in this section, we will investigate the anisotropic VB soliton dynamics by applying a kick to a stationary soliton solution, where k x(y) is the kick. According to Ref. [50], the trajectory of the moving anisotropic VB soliton also can be defined by time-dependent coordinates of the center of mass of the bright component  Fig. 4. When the kick strength is small, the resulting trajectory is a butterfly-shaped, which means the mobility of the kicked anisotropic VB soliton is Rabbi-like oscillation. The strong kick only induces the larger oscillation amplitude, as shown in Fig. 4b. We then examine the two mean displacements in x and y directions For clarity, the corresponding drifts X and Y versus time t are given in Fig. 4c and d. We can see that the mobility of the kicked anisotropic VB soliton obviously is Rabbi-like oscillation.
For the 2D anisotropic case, collapse is an important issue. Then, we also consider the collision dynamics of two anisotropic VB solitons. To do this, we simulate the Eq. (1) and the initial state was taken as two oppositely moving anisotropic VB solitons of the form where L and R represent the two different anisotropic VB solitons on the left and right. This ansatz approximates a solution comprising two anisotropic VB solitons located at −(x 0 , y 0 ) and (x 0 , y 0 ). For simplicity, we choose (x 0 , y 0 ) = (x, 0), which means two anisotropic VB solitons initially place at the x axis. For the case in free space, it has been demonstrated that two anisotropic VB solitons with small initial distance may merge after inelastically colliding and they do not collide each other when they have big initial distance. For the case in the presence of OL, they always merge after inelastically colliding and can induce the transition from the bright component to the vortex component depending on the big initial distance (x 0 , y 0 ).

Conclusion
In summary, we have studied construction and dynamics of stable 2D anisotropic VB soliton in the model of the spinor BECs with the SOC and DDI confined in a 2D OL. It is found that the OL and DDI can support stable anisotropic VB soliton in the present setting for arbitrarily small value of N , which is different from the case without OL in Ref. [50]. We then present a new method via examining the mean square error of norm share of bright component to implement stability analysis. It is revealed that one can control the stability of anisotropic VB soliton only by adjusting OL depth V 0 for a fixed DDI strength. In addition, the dynamics of the anisotropic VB soliton was studied by applying the kick to them. The mobility of the single kicked anisotropic VB soliton is Rabbi-like oscillation. However, for the collision dynamics of two kicked anisotropic VB solitons, their properties mainly depend on the OL and initial distance (x 0 , y 0 ). For the case in free space, it has been demonstrated that two anisotropic VB solitons with small initial distance may merge after inelastically colliding and they do not collide each other when they have big initial distance. For the case in the presence of OL, they always merge after inelastically colliding and can induce the transition from the bright component to the vortex component depending on the big initial distance. Our results may provide a convenient way to prepare and manipulate anisotropic VB soliton in high-dimensional space. With currently available techniques, it is possible to realize our model and observe our theoretical predictions with experiments. Our proposed two-component (spinor) BECs with the linear SOC between the components can be demonstrated experimentally in coldatom setups [54]. For instance, one can use two pairs of counter-propagating laser beams with orthogonal polarizations to realize 2D OL [55][56][57]. Long-range DDI of atoms can be introduced and tuned by using a rotating polarizing magnetic field [58][59][60][61]. For such a two-component BECs with the SOC and DDI confined in a 2D OL, stable anisotropic VB soliton may be observed. In addition, our approach may also be used to explore stable anisotropic VB soliton in the 3D setting with SOC and DDI. ent project of Central South University of Forestry and Technology under Grant No. 2017YJ035.

Funding
The authors have not disclosed any funding.
Data availability All data generated or analyzed during this study are included in this published article.

Conflict of interest
The authors declare that they have no conflict of interest.
Appendix A: Equivalence of the mean square errors of F 1 and F 2 In Eq. (13), we use the mean square error of norm share of bright component as a criterion to implement stability analysis. Here, we will prove the mean square errors of F 1 and F 2 are equivalent. According to the definition, the mean square error of norm share of vortex component defined as where the time-dependent norm share of the vortex component F 2 (t) = |ψ 2 (x, y, t)| 2 dxdy/N , the average value of the norm share of the vortex component in the evolution F 2 = F 2 (t) and the variance F 2 (t) = F 2 (t)− F 2 . Because our system is close, we have F 1 (t) + F 2 (t) = 1 and F 1 + F 2 = 1. Combining the Eq. (13), we can obtain Obviously, the mean square errors of F 1 and F 2 are equivalent, 2 F 1 (t) = 2 F 2 (t) .