Bubbles and W-shaped solitons in Kerr media with fractional diffraction

We demonstrate that, with the help of a Gaussian potential barrier, dark modes in the form of a local depression (“bubbles”) can be supported by the repulsive Kerr nonlinearity in combination with fractional dimension. Similarly, W-shaped modes are supported by a double potential barrier. Families of the modes are constructed in a numerical form, and also by means of the Thomas–Fermi and variational approximations. All these modes are stable, which is predicted by computation of eigenvalues for small perturbations and confirmed by direct numerical simulations.