Stable three-dimensional solitons in spin-orbit-coupled atomic-molecular condensates

We elaborate a mechanism for the creation of stable three-dimensional (3D) solitons in spin-orbit-coupled (SOC) atomic-molecular Bose-Einstein condensate, modeled by the mean-field equations with the quadratic three-wave interaction, characterized by mismatch $\alpha $. The planar (effectively two-dimensional) SOC is applied to the soliton's atomic component, structuring it as a mixed mode (MM) or semi-vortex (SV). The molecular component of the SV soliton is shaped as a 3D vortex, while the molecular component in the MM soliton is an MM too. The solitons exist up to a critical value of $\alpha $. The system demonstrates a relatively large norm share of the vortex components, exceeding $50\%$ of the total norm, which is an essential feature of SOC-supported solitons. This is scheme for realizing stable vortex solitons in free space with the quadratic nonlinearity.