Gap Solitons in Fractional Dimensions With a Quasi-Periodic Lattice

The existence and stability of gap solitons in the nonlinear fractional Schrodinger equation are investigated with a quasi-periodic lattice. In the absence of nonlinearity, the exact band-gap spectrum of the proposed system is obtained, and it is found that the spectrum gap size can be adjusted by the sublattice depth and the Levy index. Under self-defocusing nonlinearity, both in-phase and out-of-phase gap solitons have been searched in the first four gaps. It is revealed that in-phase gap solitons are generally stable in wide regions of their existence, whereas stable out-of-phase gap solitons can only exist in the fourth spectrum gap. Linear stability analysis of gap solitons is in good agreement with their corresponding nonlinear evolutions in fractional dimensions. The presented numerical findings may lead to interesting applications, such as transporting of light beams through the optical medium, and other areas connected with the Kerr effect and fractional effect.