Fractional-order effect on the stability of the coexistence state of ring dark and anti-dark solitons with vortex

Vortex plays a certain stabilizing role for the ring in the coexistence state of ring dark and anti-dark solitons with vortex due to the interaction between solitons and vortex in Bose-Einstein condensation and nonlinear optics. The numerical results show that the Lévy index for the Fractional-order effect has a great influence on their propagation. When the Lévy index is the optimal value, the ring dark soliton can maintain stable transmission over a longer distance, and the ring anti-dark soliton can also maintain stable transmission over a certain distance. •The coexistence state of ring dark and anti-dark solitons with vortex is revealed.•Fractional-order effect on the stability of this coexistence state is studied.•The coexistence states in both Kerr nonlinear medium and saturable nonlinear medium are reported.