An effective equation for quasi-one-dimensional funnel-shaped Bose–Einstein condensates with embedded vorticity

On the basis of a recently introduced model for the Bose–Einstein condensate (BEC) trapped in the 2D “funnel” potential, ∼ - r^{- 1}, we develop analysis for vortex modes, which are confined in the transverse direction by the self-attraction, or by the trapping potential, in the case of self-repulsion. Linear 3D wave functions are found exactly for eigenstates with an orbital momentum. In the case of self-repulsion, 3D wave functions are obtained by means of the Thomas–Fermi approximation. Then, with the help of the variational method, the underlying Gross–Pitaevskii equation is reduced to a 1D nonpolynomial Schrödinger equation (NPSE) for modes with zero or nonzero embedded vorticity, which are tightly confined by the funnel potential in the transverse plane. Numerical results demonstrate high accuracy of the NPSE reduction for both signs of the nonlinearity. The analysis is performed for stationary modes and for traveling ones colliding with a potential barrier. By means of simulations of NPSE with the self-attraction, collisions between solitons are studied too, demonstrating elastic and inelastic outcomes, depending on the impact velocity and underlying vorticity. A boundary of the stability of 3D vortices with winding number S= 1 against spontaneous splitting in two fragments is identified in the case of the self-attraction, all vortices with S≥ 2 being unstable.