Interactions of fractional solitons with local defects: Stabilization and scattering

TL;DR

This paper investigates the stabilization and scattering of fractional solitons in nonlinear media with fractional diffraction, demonstrating defect pinning as an effective stabilization method and analyzing soliton interactions with defects.

Contribution

It introduces a method to stabilize fractional solitons using delta-functional defects and provides analytical and numerical analysis of their behavior in various fractional regimes.

Findings

Fractional solitons are stabilized by defect pinning at specific Levy indices.

The Vakhitov-Kolokolov criterion accurately predicts instability boundaries.

Collisions of solitons with defects result in rebound, splitting, or passage.

Abstract

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 Levy index (LI) taking values LI < 2. Fractional solitons are unstable at LI smaller or equal to 1, or LI smaller or equal to 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 Schroedinger 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 LI = 1, and partly stabilized for LI < 1. In the case of the quintic self-focusing, the full and partial stabilization are found for LI = 2 and LI < 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.