Dark soliton solution of the three-dimensional Gross-Pitaevskii equation with an isotropic harmonic potential and nonlinearity in polytropic approximation

We study the three-dimensional Fermi gas in an isotropic harmonic trap during the Bardeen- Cooper-Schrieffer superfluid to Bose-Einstein condensate (BCS-BEC) crossover, which is modeled by using the generalized Gross-Pitaevskii equation (GGPE) in the polytropic approximation. We analytically solved the 3D GGPE with a coupled modulus-phase transformation without introducing any additional integrability constraint, reaching the dark soliton-like solution. We find that the dark soliton identified undergoes an oscillation with a constant period over the whole BCS-BEC crossover region, although the amplitude of the dark soliton varies with polytropic index, demonstrating the peculiar nonlinear properties for the system modeled by using the 3D GGPE.