Creating solitons by means of spin-orbit coupling

This mini-review collects theoretical results predicting the creation of matter-wave solitons by the pseudo-spinor system of Gross-Pitaevskii equations (GPEs) with the self-attractive cubic nonlinearity and linear first-order-derivative terms accounting for the spin-orbit coupling (SOC). In one dimension (1D), the so predicted bright solitons are similar to their well-known counterparts supported by the GPE in the absence of SOC. Completely novel results were recently obtained for 2D and 3D systems: SOC suppresses the collapse instability of the multidimensional GPE, creating fully stable 2D ground-state solitons and metastable 3D ones of two types: semi-vortices (SVs), with vorticities m = 1 in one spin component and m = 0 in the other, and mixed modes (MMs), with m = 0 and m = (+/-)1 present in both components. With the Galilean invariance broken by SOC, moving solitons exist up to a certain critical velocity, suffering delocalization above it. The newest result predicts stable 2D "quantum droplets" of the MM type in the presence of the Lee-Huang-Yang corrections to the GPE system, induced by quantum fluctuations around the mean-field states, in the case when the inter-component attraction dominates over the self-repulsion in each component.