Motion of ferrodark solitons in harmonically trapped superfluids: spin corrections and emergent quartic potentials exhibiting symmetry breaking

We propose a framework for topological soliton dynamics in trapped spinor superfluids, decomposing the force acting on the soliton by the surrounding fluid into the buoyancy force and spin-corrections arising from the density depletion at soliton core and the coupling between the orbital motion and the spin mixing, respectively. For ferrodark solitons (FDSs) in spin-1 Bose-Einstein Condensates (BECs), the spin correction enables mapping the FDS motion in a harmonic trap to the atomic-mass particle dynamics in an emergent quartic potential. Initially placing a type-I FDS near the trap center, a single-sided oscillation happens, which maps to the particle moving around a local minimum of the emergent double-well potential. As the initial distance of a type-II FDS from the trap center increases, the motion exhibits three regimes: trap-centered harmonic and anharmonic, followed by single-sided oscillations. Correspondingly the emergent quartic potential undergoes symmetry breaking, transitioning from a single minimum to a double-well shape, where particle motion shifts from oscillating around the single minimum to crossing between two minima via the local maximum, then the motion around one of the two minima. In a hard-wall trap with linear potential, the FDS motion maps to a harmonic oscillator.