We demonstrate that higher-order vortex solitons (with larger vortex charges m) in the nonlinear Schrödinger equation with cubic (defocusing Kerr) nonlinearity can be stabilized by the action of a flat-bottom potential. Such a model can also support the common fundamental (vorticityless) solitons, including the Gaussian-like and flat-top solitons. The effective radii of the fundamental and vortex solitons can be modified by tuning the radius of the flat-bottom part of the potential. We display the contours and phases of vortex solitons with vorticity numbers up to \(m=12\) . Interestingly, the central holes of vortex solitons increase with the increase of vortex charges, and the central portions of these higher-order vortices become flat. We investigate the existence and stability of both the fundamental and vortex solitons, for different values of the relevant model parameters: the radius of the flat bottom, initial beam power, propagation constant, and the strength of nonlinearity. We also consider the propagation of perturbed initial beams, as well as the propagation in longitudinally modulated flat-bottom potentials. Such a modulated propagation allows for an easy soliton management. We find that the fundamental solitons are completely stable, while the higher-order vortex solitons are prone to the modulation instability, degenerating into m simple vortices that fly away from the inside to the outside of solitons. Eventually, an initial higher-order vortex beam turns into a fundamental (chargeless) soliton.
