Defect solitons supported by nonlinear fractional Schrödinger equation with a defective lattice

We investigate numerically the existence and stability of defect solitons in nonlinear fractional Schrödinger equation. For positive defects, defect solitons are only existent in the semi-infinite gap and are stable in their whole existence domain irrespective of Lévy index. For moderate deep defects, defect solitons are existent in both the semi-infinite gap and first gap, and their instability domains occur in the low-power region of the semi-infinite gap. While for deep enough defects, stable defect solitons can be found in the second gap. Increasing the strength of defect (or Lévy index) will narrow (or broaden) the existence and stability domains.