Cavity-assisted matter-wave solitons in a spin–orbit-coupled Bose–Einstein condensate

We investigate nonlinear matter-wave solitons supported by a spin–orbit-coupled (SOC) Bose–Einstein condensate (BEC) confined inside a high-Q Fabry–Pérot cavity. Starting from the cavity-QED description of a Raman-dressed two-component condensate, we formulate the semiclassical dynamics via quantum Langevin equations and connect the driven cavity-SOC system to a soliton-compatible nonlinear evolution model by adiabatically eliminating the intracavity field and projecting onto dressed atomic branches. In the conservative limit considered here, we show that the resulting effective envelope dynamics reduces to a scalar nonlinear Schrödinger-type equation whose exact Madelung transformation yields a real two-field hydrodynamic system. Using a traveling-wave reduction, we obtain an ordinary differential equation governing the soliton profile and construct several families of exact solutions via a modified F-expansion method. The resulting waveforms include localized dark-type notches on a constant background, sharply peaked (spike-like) excitations, and periodic soliton-train (soliton-lattice) patterns, all exhibiting shape-preserving propagation with a velocity fixed by the traveling coordinate. Although the exact construction is performed for a single envelope field, we demonstrate that the solutions correspond to SOC spinor solitons in the physical pseudo-spin basis: the two matter-wave components are reconstructed from the same envelope with weights determined by the Raman/Zeeman mixing angle, yielding tunable pseudo-spin compositions. These analytical soliton solutions provide a baseline for understanding nonlinear localization in cavity-coupled SOC quantum gases and establish a platform for future extensions to driven-dissipative regimes and cavity-enabled photonic responses.