# Two-Dimensional Vortex Solitons in Spin-Orbit-Coupled Dipolar Bose-Einstein Condensates
## 1 Introduction  
Creating stable solitons or self-trapped modes in systems with a dimension higher than one remains a challenging and attractive problem in physics, both in theory and experiments [1; 2]. The main obstacle lies in the ubiquitous cubic nonlinearity, which could be the Kerr effect in optics [3] or the intrinsic ones induced by two-body scattering in Bose-Einstein condensates (BECs) [4; 5; 6]. Although these nonlinearities can support stable soliton-like modes in 1D geometry, in high dimensions like 2D or 3D systems, the collapse driven by cubic self-focusing nonlinearities will destroy the stable modes [7; 8; 9]. Several methods have been developed, theoretically and experimentally, to solve this problem. One is using alternative nonlinearities other than cubic, such as quadratic self-focusing nonlinearity, which can support stable 2D spatial solitons [10] and spatiotemporal soliton chains [11] in optics, mixed cubic-quintic or quadratic-cubic ones, which can support stable fundamental spatial solitons [12; 13; 14; 15; 16]. The other is introducing spatially periodic potentials, which is known to be able to help stabilize 2D and even 3D fundamental and vortical solitons in plentiful settings [17; 18; 19; 20], and thespectral band these periodical potentials host can create gap solitons, as well [17; 21; 22; 23; 24]. Another method is to use nonlocal nonlinearities, which can came from the long-range interaction between the molecules in liquid crystals [25], the diffusive behavior in nonlinear medium [26; 27], the van der Waals interactions between Rydberg atoms formed in BEC [28], the dipole-dipole interaction (DDI) and quadrupolar interaction between the permanent [29; 30] or the external fields' induced [31; 32; 33; 34; 35; 36] magnetic or electric dipole moments of the BEC atoms. Besides the above methods, there is a new approach recently found theoretically by using a two-component BEC with linear spin-orbit coupling (SOC) between the two components. Combined with repulsive nonlinearity, such a system is known to support 1D and 2D bright solitons [37; 38], 2D dark vortices [39; 40; 41; 42] and bright gap solitons (with an optical lattice) [43; 44; 45; 46]. With intrinsic attractive nonlinearity, BEC with SOC was thought to be able to support 1D solitons only [37; 47; 48], but new work revealed, very surprisingly, that with Rashba-type SOC (see Equation (2)), BEC with attractive cubic nonlinearity can support stable 2D semi- and mixed-mode vortex solitons in free space [49; 50; 51; 52; 53]. A similar mechanism can also be established with optical settings [54; 55], and the excited state soliton can be created by introducing the inverted nonlinear potential [56; 57]. Combined with long-range interactions, dipolar BEC with SOC is found to support 2D bright solitons, as well, which could be a plane-wave soliton with a spatially varying phase or a stripe soliton with a spatially oscillating density for each component [38]. These findings not only open new possibilities to create stable multi-dimensional solitons, but can be used to investigate 2D or 3D solitons with vortices and other topological properties, as well. In this article, we reviewed recent theoretical results of creating stable 2D solitons in free space by using BEC with SOC and DDI nonlinearity. In the setting of SOC BEC with pure DDI interactions, families of stable symmetric semivortices (SVS) are found, and the properties of these SVS families are quantified [58]. Families of stable asymmetric semivortices (AVS) are also found in this setting, which are absent in systems with contact interactions [58]. Control parameters and stable conditions of these AVS families are discussed and analyzed [58]. Both families of soliton are anisotropic in space due to the anisotropic nature of the DDI. By introducing a Zeeman splitting (ZS) into this setting, it is found that the shape of the SVS can be controlled. By increasing the strength of the Zeeman splitting, a transition from horizontal SVS to vertical SVS occurs [59].