Motion of Ferrodark Solitons in Trapped Superfluids: Spin Corrections and Emergent Oscillators

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. Our formulation applies to large-amplitude soliton motion in general superfluids with spin degrees of freedom under arbitrary external potentials. For ferrodark solitons (FDSs) in spin-1 Bose-Einstein condensates, the spin correction could diverge, change the direction of the total force, and enable 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 oscillations followed by single-sided oscillations. Correspondingly the emergent quartic potential undergoes a transition from a single minimum to a double-well shape, where the particle motion shifts from oscillating around the single minimum to crossing between two minima via the local maximum, then the symmetry-breaking motion around one of the two minima. In a hard-wall trap with linear potential, the FDS motion maps to a harmonic oscillator.