Interactions of fractional solitons with local defects: Stabilization and scattering

Stability is an essential problem in theoretical and experimental studies of solitons in nonlinear media with fractional diffraction, which is represented by the Riesz derivative with Lévy index (LI) $α$⁠, taking values $α<2$⁠. Fractional solitons are unstable at $α≤1$ or $α≤2$ in uniform one-dimensional media with the cubic or quintic self-focusing, respectively. We demonstrate that, in these cases, the solitons may be effectively stabilized by pinning to a delta-functional trapping potential (attractive defect), which is a relevant setting in optical waveguides with the effective fractional diffraction. Using the respective fractional nonlinear Schrödinger equation with the delta-functional potential term, we find that, in the case of the cubic self-focusing, the fractional solitons are fully stabilized by the pinning to the defect for $α=1$ and partly stabilized for $α<1$⁠. In the case of the quintic self-focusing, the full and partial stabilization are found for $α=2$ and $α<2$⁠, respectively. In both cases, the instability boundary is exactly predicted by the Vakhitov–Kolokolov criterion. Unstable solitons spontaneously transform into oscillating breathers. A variational approximation (VA) is elaborated parallel to the numerical analysis, with a conclusion that the VA produces accurate results for lower LI values, i.e., stronger fractionality. In the cubic medium, collisions of traveling stable solitons with repulsive and attractive defects are addressed too, demonstrating outcomes in the form of rebound, splitting, and passage.