Bright solitons under fractional diffraction and various harmonic potentials

We demonstrate that bright solitons can be sustained in various harmonic potentials under fractional diffraction. In addition to the parabolic shape, the potentials include an added constant strength. By varying that strength, we examine soliton profiles, their powers, and stability regions. The power curves of these solitons as functions of the Lévy index and the propagation constant are also presented, illustrating significant changes. The stability domains are explored by the linear stability analysis and verified by direct numerical simulations. We find that all quantities describing solitons are highly sensitive to variations in the three relevant parameters: the constant strength, the Lévy index, and the propagation constant. Interestingly, the instability of solitons is promoted by an increase in the constant strength. Finally, examples of both stable and unstable propagations of perturbed solitons are depicted, for different values of parameters.