Three-dimensional solitons in dipolar coupled atomic–molecular Bose–Einstein condensates with PT-symmetric photonic lattices

We explore the formation and properties of three-dimensional (3D) matter-wave solitons in atomic–molecular Bose–Einstein condensates (AMBECs) with dipole–dipole interactions (DDI) under the action of parity–time (PT)-symmetric photonic lattices. The model incorporates quadratic nonlinearity, which coherently couples atomic and molecular components. Stationary soliton solutions are obtained by means of numerical methods. In the isotropic DDI regime, we identify stable ring-shaped vortex solitons (VSs) and quadrupole solitons with topological charges up to m=2. When DDI anisotropy is introduced, ring-shaped VSs break the symmetry, splitting into multi-core configurations, such as 2-core and 4-core ones, only VSs with m=1 remaining stable in that case. A systematic analysis reveals that the size, stability, and chemical potential of the solitons can be efficiently tuned by varying key parameters of the system, such as the DDI strength, atomic–molecular coupling coefficient, and the strength of the cubic nonlinearity. These results demonstrate rich soliton physics in dipolar AMBECs and provide a pathway for manipulating structured matter waves by means of inter-component coupling and nonlocal interactions.